Willmore and Generalized Willmore Energies in Space Forms Thanuja

Willmore and Generalized Willmore energies in space forms

Thanuja Paragoda Gamage, M.Sc.

A Dissertation

Mathematics and Statistics

Submitted to the Graduate Faculty of Texas Tech University in Partial Fulﬁllment of the Requirements for the Degree of

DOCTOR OF PHILOSOPHY

Approved

Dr. Magdalena Toda Co-Chairperson of the Committee

Dr. Giorgio Bornia Co-Chairperson of the Committee

Dr. Eugenio Aulisa

Dr. Mark Sheridan Dean of the Graduate School

August, 2016 c 2016, Thanuja Paragoda Texas Tech University, Thanuja Paragoda, May 2016

ACKNOWLEDGMENTS

First and foremost I would like to express my profound appreciation and gratitude to my advisors, Dr. Magdalena Toda, Dr. Giorgio Bornia and Dr. Eugenio Aulisa who introduced this problem to me and guided and encouraged me in every way. They continuously helped me by offering advice, guidance and instruc- tions. Without their help, this dissertation would not have been possible. Since the day that I applied to Texas Tech University for my graduate studies, they have helped me in numerous ways. Since the moment I was first confronted with dif- ficulties in doing research, they have been like a lovely family to me. I am also grateful to Dr. Bhagya Athukorallage for helping me with several numerical cal- culations and graphs in this thesis. In addition, my heartfelt gratitude goes to Texas Tech University, Texas government and United States of America for select- ing me and funding my education. I fondly remember my teachers who gave me a wonderful education during the past five years at Texas Tech and during my entire student life. I would also like to acknowledge Dr. Kumudu Mallawarachchi and other faculty members from the Department of Mathematics, University of Kelaniya, Sri Lanka. I would like to thank my Sri Lankan friends in Lubbock who were with me during the most difficult situations. Last, but not least, I would like to thank my loving parents, sisters and friends in Sri Lanka, USA and elsewhere for their support and encouragement.

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TABLE OF CONTENTS

ACKNOWLEDGMENTS ...... ii ABSTRACT ...... v LIST OF TABLES ...... vii LIST OF FIGURES ...... viii 1. INTRODUCTION ...... 1 2. FUNDAMENTAL CONCEPTS IN DIFFERENTIAL GEOMETRY AND CALCULUS OF VARIATIONS ...... 3 2.1 Introductory notions on differentiable manifolds ...... 3 2.2 Riemannian metric ...... 7 2.2.1 Operators defined on Riemannian manifolds ...... 10 2.3 Two-dimensional Riemannian manifolds ...... 12 2.3.1 First and second fundamental forms ...... 12 2.3.2 Mean and Gauss curvatures ...... 15 2.4 Variations of a functional ...... 19 3. WILLMORE ENERGIES ...... 21 3.1 Definition of Willmore energy ...... 21 3.2 The Euler-Lagrange equation of the Willmore functional . . . 23 4. GENERALIZED WILLMORE ENERGIES ...... 26 4.1 Definition of Generalized Willmore energy ...... 26 4.2 Generalized Willmore flow equation ...... 27 4.3 Generalized Willmore equation on a graph ...... 30 5. CLASSICAL AND DEFORMED WILLMORE ENERGIES IN SPACE FORMS ...... 34 5.1 Sectional curvature and Space forms ...... 34 5.2 Intrinsic versus extrinsic curvatures in space forms ...... 35 5.3 Deformed Willmore energies in Euclidean space ...... 37 5.4 Classical Willmore energies in space forms and their Euler- Lagrange equations ...... 38 5.5 Comparative Analysis ...... 44

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5.6 Deformed Willmore energies in space forms and their Euler- Lagrange equations ...... 45 6. NUMERICAL RESULTS ...... 46 6.1 Automatic Differentiation (AD) for exact Jacobian computation of the PDE systems ...... 46 6.2 Time dependent solutions of the Generalized Willmore equation on a general surface: Dirichlet boundary conditions . . . 47 6.3 Computational models of the Clifford torus ...... 48 6.4 Steady-state graph solutions of the Willmore equation: Dirich- let boundary conditions ...... 49 6.4.1 Finite element weak formulation ...... 49 6.4.2 The sphere as a Steady-State Solution of the Willmore flow 52 6.4.3 The Clifford torus as a Steady-State Solution of the Will- more flow ...... 56 6.4.4 Generalized Willmore Torus ...... 58 6.4.5 The Catenoid as a Steady-State Solution of the Willmore flow...... 60 6.5 Steady-state graph solutions of the Willmore equation: Dirichlet- Neumann boundary conditions ...... 63 6.5.1 Finite element weak formulation ...... 63 6.5.2 The Clifford torus as a Steady-State Solution of the Will- more flow with Dirichlet-Neumann boundary conditions . 65 6.6 The disk of genus of two as a steady-state graph solution of the Willmore flow ...... 69 7. CONCLUSIONS ...... 73 BIBLIOGRAPHY ...... 74 8. Appendix: Computer code ...... 77

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ABSTRACT

This PhD dissertation consists of two sections. In the first part of this work, we study a Generalized Willmore flow equation and its numerical applications. First, we derive the time dependent equation which relates to the geometric evolution of a Generalized Willmore flow on a general immersion. Then, we transform this equation into a coupled system of second order nonlinear PDEs together with the mean curvature formula on a general immersion in R3. The important point is that the Gauss curvature appears in terms of the second order derivatives in our coupled system. Moreover, we study finite element numerical solutions for steady-state cases obtained with the help of the FEMuS (Finite Element Mul- tiphysics Solver) library. We use automatic differentiation (AD) tools to compute the exact Jacobian of the coupled PDE system subject to Dirichlet boundary conditions. Next, we study the Generalized Willmore flow in the graph case. This equation is reformulated in divergence form as a coupled system of second order nonlinear PDEs together with the mean curvature formula for the graph case. In fact, the Gauss curvature does not appear in our coupled system for the graph case. Furthermore, we study finite element numerical solutions for steady-state cases and use AD tools to compute the exact Jacobian of the coupled PDE system subject to different boundary conditions such as Dirichlet and Dirichlet-Neumann boundary conditions. This is a novel method - never used previously on computing/constructing Willmore surfaces. We study the accuracy of the algorithm by providing nontrivial steady-state numerical solutions to compute the Jacobian in the Newton linearization of the finite element weak formulation. In the second part, we study Willmore-type energies and Willmore-type immersions in space forms. In fact, we introduce the notion of deformed Willmore 3 ¯ R 2 energy for a space form M (k0) as W = M (H + k1) dS such that the constants k0, k1 are independent. Next, we discuss the corresponding Euler-Lagrange equation for the deformed energy. Its novel contributions include the following:

• deriving the Euler-Lagrange equation corresponding to the classical Will- more functional if we consider surfaces immersed in a general ambient space 3 form M (k0) of constant sectional curvature k0. In addition, we deduce the

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Euler-Lagrange equation of the deformed Willmore energy in a space form, in a uniﬁed way, using an extrinsic Laplace-Beltrami operator (which depends on both the surface metric, and the ambient space form). We consider both the case of closed surfaces and the one of surfaces with boundary, for which we gave and discussed the necessary boundary value conditions, which the previous literature failed to do;

• determining the natural choices of the constant k1 in the deformed Willmore functional such that the corresponding the Euler-Lagrange equation which is given by 2 ∆gH + 2H(H − K) = 0.

This is the same form that is obtained for the classical Willmore functional in the Euclidean ambient space.

Thus, this dissertation provides a bridge between prior works in the ﬁeld, as well as a novel approach both from a purely mathematical point of view and in the context of applications.

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LIST OF TABLES

6.1 L2 error and order of convergence for spherical cap ...... 54 6.2 Seminorm error and order of convergence for spherical cap ...... 55 6.3 L2 error and order of convergence for Clifford torus ...... 57 6.4 Seminorm error and order of convergence for Clifford torus . . . . . 58 6.5 L2 error and order of convergence for Catenoid ...... 62 6.6 Seminorm error and order of convergence for Catenoid ...... 62 6.7 L2 error and order of convergence for Clifford torus with Dirichlet- Neumann boundary conditions ...... 66 6.8 Seminorm error and order of convergence for Clifford torus with Dirichlet-Neumann boundary conditions ...... 67

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LIST OF FIGURES

2.1 Normal curvature kn and principal curvatures k1 and k2 at point P on curved surface...... 15 6.1 The domain in the xy-plane of the Clifford torus ...... 49 6.2 The X-profile of the Clifford torus with = 0 ...... 50 6.3 The Y -profile of the Clifford torus with = 0 ...... 51 6.4 The Z-profile of the Clifford torus with = 0 ...... 52 6.5 The mean curvature of the Clifford torus with = 0 ...... 53 6.6 The sector of a spherical cap profile U with = 0 ...... 55 6.7 The sector of spherical cap weighted mean curvature W with = 0 . 56 6.8 On the left: cross section of some computed Generalized Willmore torus surfaces for the angle range π/6 ≤ u ≤ 5π/6, for various values of . The dashed curve is the Clifford torus. The curves for = 0.03 (A), = 0.01 (B) and = 0.001 (C) are visible on the right, in a zoom of the dash-dotted rectangle...... 59 6.9 Generalized Willmore torus profile U with = 0.01 ...... 59 6.10 Generalized Willmore torus weighted curvature W with = 0.01 .. 60 6.11 Catenoid profile U ...... 63 6.12 Weighted mean curvature W of the catenoid ...... 64 6.13 Clifford torus profile U subject to the Dirichlet-Neumann boundary conditions ...... 67 6.14 Weighted mean curvature W subject to the Dirichlet-Neumann boundary conditions ...... 68 6.15 Dirichlet-Neumann boundary conditions for U ...... 68 6.16 Dirichlet-Neumann boundary conditions for W ...... 69 6.17 Disk with two holes ...... 70 6.18 Profile of a disk with two holes ...... 71 6.19 The weighted mean curvature ...... 72

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CHAPTER 1 INTRODUCTION

The study of Willmore energies is a fast-developing field with a large array of mathematical, physical and biological applications. This work consists of studying some generalizations of Willmore energy in space forms. Let us summarize the main contents of the various chapters. In Chapter 2, we provide a short sur- vey of some crucial fundamental concepts in differential geometry and calculus of variations. We will use these basic notions in other chapters of this dissertation. The definitions we provide here can be found in many fundamental textbooks of differential geometry. Chapter 3 illustrates an introduction to Willmore energies and the corresponding Euler-Lagrange equation. We give a description of how the Willmore energy is defined, related theorems and minimizers of the Willmore energy. Chapter 4 addresses Generalized Willmore energy on an immersed surface in R3. This Generalized Willmore energy (sometimes called deformed Willmore energy) is a composition of classical Willmore energy and the energy due to the surface tension. In the literature, some authors considered the different versions of Generalized Willmore energies such as Helfrich energy for lipid bilayers. In fact, there are new models of Generalized Willmore surfaces (critical points of Gener- alized Willmore energy), for instance in structural mechanics, molecular biology and biophysics. The beta barrels in protein structures can be interpreted as various rotational Generalized Willmore surfaces with no self intersections. In the past four decades, many authors studied geometric evolution equations, mean curvature flow and Willmore flow both from a theoretical point of view and a numerical one. In this chapter, we study a Generalized Willmore flow equation on an immersed surface in R3 and for a graph particular. In both cases, we remodeled the Generalized Willmore flow equation into coupled PDE systems subject to different boundary value conditions. Chapter 5 yields classical and deformed (generalized ) Willmore energies in space forms. We begin the study of sectional curvature, space forms and intrinsic versus extrinsic curvatures in space forms. Furthermore, we prove the Euler-

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Lagrange equation corresponding to the classical Willmore functional in a general ambient space which brings a novel approach in the study of Willmore energies in space forms. The important point of our proofs is that we introduce the concept of the extrinsic Laplace-Beltrami operator which depends on both the surface metric, and the ambient space form. In the last part of this chapter, we provide a sum- mary of comparisons between our work and other studies. One of the main goals of this work is to provide computational models of generalized Willmore surfaces using Automatic Differentiation (AD) and finite element methods. In Chapter 6, we study finite element numerical solutions of our coupled PDE systems which we obtained in the Chapter 4. We use AD techniques for exact Jacobian computation of the PDE systems since the classical methods such as numerical differentiation and symbolic differentiation lead us problems with calculating higher derivatives and the increase of complexity and errors. First, we study time dependent solutions of the Generalized Willmore equation on a general surface subject to the Dirichlet boundary conditions. We compare our numerical solutions with known analytical Clifford torus solutions and some color maps of the Clifford torus are presented. Next, we investigate steady-state graph solutions of the Willmore equation subject to Dirichlet boundary conditions and Dirichlet-Neumann boundary conditions. We test our algorithm using known analytical solutions such as a spherical cap and a sector of a Clifford torus which are the solutions of Willmore equation in the graph case. Moreover, we study L2 errors, semi-norm errors and orders of convergence of our coupled PDE system in the graph case. Next, we solve the Generalized Willmore equation and construct a Generalized Willmore torus which is a deformation of the actual Clifford torus and show color maps of the profile and the weighted mean curvature of the Generalized Willmore torus. It was recently shown that any Willmore graph on a bounded C4 domain Ω with vanishing mean curvature on the boundary ∂Ω must be a minimal graph (see [24] for further details). We compare our numerical Willmore graph solutions with known analytical solutions and then present the convergence results and visual representations.

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CHAPTER 2 FUNDAMENTAL CONCEPTS IN DIFFERENTIAL GEOMETRY AND CALCULUS OF VARIATIONS

In this chapter, we provide a brief introduction to the fundamental concepts in general manifold theory with the speciﬁcation of the two-dimensional case which is needed as a background material for the core of this thesis. They can be found in most textbooks of differential geometry [17, 25, 36].

2.1 Introductory notions on differentiable manifolds Differential geometry is related to the study of geometric objects defined on differentiable manifolds. A differentiable manifold is a type of manifold which is locally almost identical to a linear space to allow one to do calculus. It is a topo- logical manifold with a globally differentiable structure which is defined below. In topology, one describes a differentiable manifold using a collection of charts which is also called an atlas. An atlas consists of individual charts that, roughly speaking, describe individual regions of the manifold. The following formal definition of a differentiable manifold can be found in [25, 26].

Deﬁnition 2.1. An n-dimensional differentiable manifold is a set M together n n with a family of injective maps fα : Uα ⊂ R → M of open sets Uα in R into M such that: S (i) fα(Uα) = M; α

−1 (ii) for each pair α, β with fα(Uα) ∩ fβ(Uβ) = W 6= ø, the sets fα (W ) and −1 n −1 −1 fβ (W ) are open sets in R and the maps fβ ◦ fα, fα ◦ fβ (sometimes called transition maps) are differentiable;

(iii) The family {(Uα, fα)} is maximal relative to (i) and (ii).

If p ∈ fα(Uα), the pair (Uα, fα) is called a parametrization (or a coordinate sys-

tem or local chart) of M at p and fα(Uα) is then called a coordinate neighborhood

of p. A family (fα,Uα) satisfying the properties (i) and (ii) is called a differentiable

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structure on M. The condition (iii) is technical, because given a differentiable structure on M, we consider the union of all the parametrizations that, together with any of the parametrizations of the given structure , satisfy condition (ii) as a maximal one [26]. From now on, when we denote a differentiable manifold by M n, the upper index stands for the dimension of the differentiable manifold.

Examples of Differentiable manifolds The following few examples of differentiable manifolds can be found in [17, 25, 26].

Example 1. Rn is a trivial example of an n-dimensional differentiable manifold with the differentiable structure given by the identity.

Example 2. A ﬁnite n-dimensional vector space V is an n-dimensional differ-

entiable manifold. In order to see that let us consider that {ei} is a basis of V and {ej} is the dual basis. Then {ej} are the coordinate functions of a global coordinate system on V , which canonically determines a differentiable structure on V .

Example 3. The n-sphere Sn is an n-dimensional differentiable manifold which is defined as n+1 n n+1 X 2 n+1 S = (x1, ..., xn+1) ∈ R ; xi = 1 ⊂ R . (2.1) i=1 Let N = (0, ..., 0, 1) and S = (0, ..., 0, −1) be the north and south poles of Sn respectively. n n n Define a mapping π1 : S − {N} → R that takes p = (x1, ..., xn+1) in S − {N} into the intersection of the hyperplane xn+1 = 0 with the line that passes through p and N. This mapping is called the “stereographic projection” from the north pole. It is easy to verify that x1 xn π1(x1, ..., xn+1) = , ..., . (2.2) 1 − xn+1 1 − xn+1 n n Similarly, we can define the stereographic projection π2 : S − {S} → R from the

south pole onto the hyperplane xn+1 = 0.

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n The mappings π1 and π2 are differentiable, injective and map S − {N} and n S − {S} onto the hyperplane xn+1 = 0. n −1 n −1 n n Hence, the parametrizations (R , π1 ), (R , π2 ) cover S . The n-sphere S is n −1 n −1 thus described by two charts and {(R , π1 ), (R , π2 )} is a differentiable structure n −1 n −1 n n on S . Observe that the intersection π1 (R ) ∩ π2 (R ) = S − {N ∪ S} is not empty. Therefore, Sn can be considered to be the simplest nontrivial example of a differentiable manifold.

n m Deﬁnition 2.2. Let M1 and M2 be differentiable manifolds. A map ϕ : M1 → m M2 is differentiable at a point p ∈ M1 if given a parametrization ρ : V ⊂ R → n M2 around ϕ(p), there exists a parametrization f : U ⊂ R → M1 around p such that ϕ(f(U)) ⊂ ρ(V ) and the map

−1 n m ρ ◦ ϕ ◦ f : U ⊂ R → R

−1 is differentiable at f (p). The map ϕ is differentiable in an open set of M1 if it is differentiable at all points of this set.

Deﬁnition 2.3. The tangent space of M at p, denoted by TpM, is deﬁned as the set of all tangent vectors which does not depend on the choice of chart.

It has been seen that at each point p of a differentiable manifold M there is an n-dimensional tangent space TpM. Therefore, the tangent spaces TpM,TqM at points p and q respectively are isomorphic since they are both n-dimensional [36].

Deﬁnition 2.4. Let M n be a differentiable manifold and let

TM = {(p, v); p ∈ M, v ∈ TpM}.

Then TM is called the tangent bundle of M.

Let M n be a differentiable manifold and at each point p ∈ M the tangent space

TpM to M at p be an n-dimensional vector space. The tangent bundle TM is the

disjoint union of all the TpM, p ∈ M, which is itself a smooth 2n-dimensional manifold.

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Deﬁnition 2.5. Let M m and N n be differentiable manifolds. A differentiable

map ϕ : M → N is an immersion if dϕp : TpM → Tϕ(p)N is injective for all p ∈ M. If, in addition, ϕ is a homeomorphism onto ϕ(M) ⊂ N, where ϕ(M) has the topology induced by N, ϕ is an embedding. If M ⊂ N and the inclusion i : M ⊂ N is an embedding, we say that M is a submanifold of N[25].

Example 1. The curve α : R → R2 defined by α(t) = (t, |t|) is not differentiable at t = 0. Example 2. The curve α : R → R2 defined by α(t) = (t3, t2) is a differentiable mapping but is not an immersion. These examples can be found in [26]. Next, we would like to describe briefly a vector field on a differentiable manifold and the concept of Lie bracket which is an important result in differentiable vector fields.

Deﬁnition 2.6. A vector ﬁeld X on M is a smooth section on the tangent bundle,

i.e., a smooth function X : M → TM such that X(p) ∈ TpM.

Let us consider a parametrization r : U ⊂ Rn → M. Then we can write

n X ∂ X(p) = ai(p) , (2.3) ∂x i=1 i ∂ where each ai : U → is a function on U and { } is the basis associated to r, R ∂xi i = 1, ..., n. According to (2.3) X is differentiable if and only if the functions ai are differentiable for some parametrization [26]. Occasionally, (2.3) is considered as a mapping X : D → F from the set D of differentiable functions on M to the set F of functions on M, which is deﬁned as

n X ∂f (Xf)(p) = ai(p) (p), (2.4) ∂x i=1 i ∀f ∈ D. Lemma 2.7. If X and Y are differentiable vector ﬁelds on a differentiable manifold M then there exists a unique vector ﬁeld Z such that, for all f ∈ D, Zf = (XY − YX)f.

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The proof of the above lemma is in [26]. Now we can deﬁne the Lie bracket.

Deﬁnition 2.8. The Lie bracket is deﬁned as

[X,Y ] = XY − YX, (2.5)

where X and Y are differentiable vector ﬁelds.

2.2 Riemannian metric Riemannian geometry is the branch of differential geometry which studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point that varies smoothly from point to point [26]. Now we deﬁne a Riemannian metric on a differentiable manifold.

Deﬁnition 2.9. A Riemannian metric on a differentiable manifold M is a corre-

spondence which associates to each point p of M an inner product h, ip (which

is a symmetric, bilinear, positive deﬁnite form) on the tangent space TpM which varies differentially in the following sense: If r : U ⊂ Rn → M is a system of coordinates around p, with

∂ r(x1, x2, ..., xn) = q ∈ r(U) and (q) = drq(0, ..., 1, ..., 0), ∂xi then ∂ ∂ (q), (q) = gij(x1, ..., xn) ∂xi ∂xj q is a differentiable function on U.

In the above deﬁnition, the function gij(= gji) is called the local representation of the Riemannian metric in the coordinate system r : U ⊂ Rn → M. A differentiable manifold with a given Riemannian metric is called a Riemannian manifold.

Deﬁnition 2.10. Let M and N be Riemannian manifolds. A diffeomorphism

f : M → N is called an isometry if hu, vip = hdfp(u), dfp(v)if(p), for all p ∈ M

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u, v ∈ TpM.

Now we provide some examples of Riemannian manifolds which can be found in [26].

n ∂ Example 1. M = R is a trivial example with and ei = (0, ..., 1, ..., 0). The ∂xi metric is given by hei, eji = δij.

Example 2. The product metric is a non-trivial example of Riemannian manifolds. If we

consider two Riemannian manifolds M1 and M2 and the cartesian product M1 ×M2 with the product structure , then π1 : M1 × M2 → M1 and π2 : M1 × M2 → M2 are

the natural projections. We deﬁne a Riemannian metric on M1 × M2 as follows:

hu, vi(p,q) = hdπ1 · u, dπ1 · vip + hdπ2 · u, dπ2 · viq,

for all (p, q) ∈ M1 × M2, u, v ∈ T(p,q)(M1 × M2). For instance, by considering the induced Riemannian metric from R2 on the circle S1 ⊂ R2 and then taking the product metric, the torus S1 × ... × S1 = T n which is called the flat torus has a Riemannian structure. Let us consider the n-dimensional Riemannian manifold M with Riemannian metric g. As in [29], “for any C∞ function ρ, the Riemannian metric g∗ = e2ρg is said to be confomal or conformally related to g. Let h be a C∞-mapping of (M, g) into another Riemannian manifold (M ∗, g∗). If the Riemannian metric h∗g∗ induced on M by h is conformal to the original metric g, then h is called a conformal mapping of (M, g) into (M ∗, g∗). (Under a conformal mapping the angle between two vectors is preserved.) h remains to be conformal under any conformal changes of metrics on M and M ∗. If h is a diffeomorphism, then h is called a conformal diffeomorphism, and (M, g) is said to be conformally diffeomorphic to (M, g∗) through h. If h is a conformal diffeomorphism of (M, g) onto itself, then h is called a conformal transformation of (M, g)”. We are going to enter to the introduction of some crucial notions such as affine connection, Riemannian connections, covariant derivative and Levi-Civita connection on a Riemannian manifold. We first consider the set of all vector fields of class C∞ on M, denoted by Σ(M) and the ring of real valued functions of class C∞ onM,

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denoted by D(M).

Definition 2.11. An affine connection ∇ on a differentiable manifold M is a ∇ mapping (denoted by (X,Y ) −→∇X Y ) ∇ : Σ(M) × Σ(M) → Σ(M) which satisfies the following properties:

(i) ∇fX+gY Z = f∇X Z + g∇Y Z

(ii) ∇X (Y + Z) = ∇X Y + ∇X Z

(iii) ∇X (fY ) = f∇X Y + X(f)Y

where X,Y,Z ∈ Σ(M) and f, g ∈ D(M).

The deﬁnition of the covariant derivative on a Riemannian manifold is included in the following theorem [26]. The covariant derivative is a way of speci- fying a derivative along tangent vectors of a manifold. It is a generalization of the directional derivative from vector calculus.

Definition 2.12. Let M be a differentiable manifold with an affine connection ∇ and a Riemannian metric h, i. A connection is said to be compatible with the metric h, i, when for any smooth curve c and any pair of parallel vector fields P and P 0 along c, we have hP,P 0i = constant.

Deﬁnition 2.13. An afﬁne connection ∇ on a smooth manifold M is called sym-

metric when ∇X Y − ∇Y X = [X,Y ] ∀X,Y ∈ Σ(M).

It is crucial to note that in a coordinate system (U, r), the fact that ∇ is sym- ∂ metric implies that ∀i, j = 1, ..., n, ∇X Xj − ∇X Xi = [Xi,Xj] = 0, Xi = . i j ∂xi We are now able to introduce the fundamental theorem of Riemannian geometry.

Theorem 2.14. (Levi-Civita). Given a Riemannian manifold M, there exists a unique afﬁne connection ∇ on M which simultaneously satisﬁes the following conditions:

a) ∇ is symmetric.

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b) ∇ is compatible with the Riemannian metric.

From now on, it is important to remark that the connection given by the above theorem will be considered as the Levi-Civita (or Riemannian) connection on M. P k k Moreover, ∇Xi Xj = k ΓijXk where Γij are called Christoffel symbols of the connection. In other words, we can deﬁne Christoffel symbols as follows:

l Deﬁnition 2.15. The Christoffel symbols Γij in terms of the metric with respect

to the local coordinates x1, x2, ...... , xn is deﬁned by l 1 lk ∂ ∂ ∂ Γij = g gjk + gki − gij . (2.6) 2 ∂xi ∂xj ∂xk

lk where gij = hXi,Xji and (g ) is the inverse of the matrix (glk).

This is a classical expression for the Christoffel symbols of the Riemannian connection in terms of the given metric [26]. Furthermore, the covariant derivative can be written in terms of Christoffel symbols as follows:

k DV X dv X k j dxi = + Γ v Xk, (2.7) dt dt ij dt k i,j which is the classical expression of the covariant derivative and it differs from the usual derivative in Euclidean space by involving the Christoffel symbols. It is k n n important to remark that Γij = 0 for the Euclidean space R . Hence, in R , the covariant derivative coincides with the usual derivative.

2.2.1 Operators defined on Riemannian manifolds In this section, we describe how operators (such as gradient, divergence, Laplace Beltrami) are defined on a Riemannian manifold. Let us assume that M is an oriented Riemannian manifold. Then the definite volume form on M in an oriented coordinate system xi is given by

p 1 n voln = |det g| dx ∧ ... ∧ dx (2.8)

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i ∂ where the dx are the 1-forms forming the dual basis to the basis vectors ∂i = ∂xi and the ∧ is the wedge product [10].

Definition 2.16. Let M be a Riemannian manifold and f ∈ D(M). The gradient of a scalar function is considered as a vector field, denoted by grad f which is defined by

hgrad f(p), vi = dfp(v), p ∈ M, v ∈ TpM, (2.9)

where dfp is the exterior derivative of the function.

In local coordinates, one has

i i i ij (∇f) = (grad f) = ∂ f = g ∂jf (2.10)

ij ij i where g is the inverse of the matrix gij such that g gjk = δk with the Kronecker i delta, δk.

Definition 2.17. The divergence of a vector field X on a Riemannian manifold M is defined as a function divX : M → R given by divX(p) = trace of the linear

mapping Y (p) → ∇Y X(p) p ∈ M.

In local coordinates, the divergence of a vector field can be written as follows: 1 p i ∇ · X = p ∂i |det g|X . (2.11) |det g| where the Einstein notation is implied such that the repeated index i is summed over [16]. In Riemannian geometry, the Laplace operator can be generalized on manifolds and this more general operator is called the Laplace Beltrami operator which is like Laplacian, and is defined as the divergence of the gradient [16, 10]. By combining the definitions of the gradient and divergence, we can define the Laplace Beltrami operator as follows:

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Deﬁnition 2.18. The Laplace Beltrami operator applied to a scalar function f in local coordinates is given by 1 p ij ∆gf = p ∂i |det g|g ∂jf . (2.12) |det g|

Note that in the case of a Euclidean manifold the Laplace-Beltrami operator reduces to the regular Laplace, namely

n X ∂2f ∆f = . ∂x 2 i=1 i

2.3 Two-dimensional Riemannian manifolds Now, we describe some notions that are speciﬁc to the case of Riemannian manifolds of dimension two.

2.3.1 First and second fundamental forms Deﬁnition 2.19. Let M be a two-dimensional Riemannian manifold and r(u1, u2) = (X(u1, u2),Y (u1, u2),Z(u1, u2)) be an immersion (of an open, simply connected disk) in R3. Then the ﬁrst fundamental form on M is

i j I = gijdu du = hdr, dri , (2.13)

where gij are called the metric coefﬁcients and these are often denoted by letters E,F, and G.

The metric g in matrix form is given by ! ! ! g11 g12 ru1 · ru1 ru1 · ru2 EF g = = = (2.14) g21 g22 ru2 · ru1 ru2 · ru2 FG

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! ! g11 g12 1 g −g g−1 = = 22 12 21 22 g g g11g22 − g12g21 −g21 g11 ! 1 G −F = (2.15) EG − F 2 −FE ! 1 G −F = (2.16) A2 −FE

where g−1 is the inverse matrix of g and √ 2 A = kru1 × ru2 k = EG − F (2.17)

Because of (2.17) the metric coefﬁcients of (2.13) satisfy

E > 0, G > 0,A2 = EG − F 2 > 0. (2.18)

These inequalities show that the metric is a positive deﬁnite quadratic form in du1, du2.

Deﬁnition 2.20. The element of area for the immersion which is denoted by dS is deﬁned as √ dS = EG − F 2 du1du2 (2.19)

Note that the property of this map r = r(u1, u2) being an immersion is equiv-

alent to the property that the vectors ru1 and ru2 are linearly independent at every point. The ﬁrst fundamental form of a surface provides information about the intrinsic geometry of the surface. It allows the calculation of curvature for instance, Gauss curvature and metric properties of a surface such as length and area in a manner consistent with the ambient space.

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Deﬁnition 2.21. The unit normal vector ﬁeld is given by

r 1 × r 2 N = u u (2.20) kru1 × ru2 k

It is sometimes called Gauss map which represents a map N : D → S2. This map provides a mapping from every point on a surface to a corresponding point on a unit sphere.

Deﬁnition 2.22. Let M ⊂ R3 be a manifold and N be a surface normal to M deﬁned in a neighborhood of a point p ∈ M. For a tangent vector v to M at p we put S(v) = −dN(v)

Then S is called the shape operator.

Note that S is a linear map S : TpM → TpM. The shape operator of a plane is identically zero at all the points of the plane. For a nonplanar surface the surface normal N can twist and turn from point to point. Therefore, S is nonzero.

Deﬁnition 2.23. The normal curvature is deﬁned as hS(v), vi κn(v) = (2.21) hv, vi

Together with the ﬁrst fundamental form, the second fundamental form serves to deﬁne extrinsic invariants of the surface, its principal curvatures. At each point p on M one may choose a unit normal vector. A normal plane at p is one that contains the normal vector, and will therefore also contain a unique direction tangent to the surface and cut the surface in a plane curve, called normal section. This curve will in general have different curvatures for different normal planes at p.

Deﬁnition 2.24. The principal curvatures at p, denoted k1 and k2, are the maximum and minimum values of the normal curvature.

These are the eigenvalues of the shape operator at p and the corresponding eigen vectors are called principal directions at that point.

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Figure 2.1. Normal curvature kn and principal curvatures k1 and k2 at point P on curved surface.

Deﬁnition 2.25. The second fundamental form on a surface is

i j II = hijdu du = −hdN, dri. (2.22)

It is considered in matrix form as follows: ! ! ! h11 h12 ru1u1 · N ru1u2 · N l m h = = = (2.23) h21 h22 ru2u1 · N ru2u2 · N m n

where hij are called the second fundamental metric coefﬁcients and these are often denoted by letters l, m, and n.

Deﬁnition 2.26. In some text books there is another way of writing the second fundamental form which is a symmetric bilinear form on the tangent plane to M at the point p and it is denoted by II(v, w) = hS(v), wi where v is a tangent vector to M at a point p and S(v) is the shape operator at that point which views as a linear map from the tangent plane to itself [17].

2.3.2 Mean and Gauss curvatures

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Deﬁnition 2.27. The arithmetic mean of the principal curvatures

1 H = (k1 + k2) (2.24) 2

is called the mean curvature of the surface.

The mean curvature H of a surface M is an extrinsic measure of curvature which locally describes the curvature of an embedded surface in some ambient space such as Euclidean space.

Deﬁnition 2.28. The surface with zero mean curvature is called minimal surface which locally minimizes its total surface area subject to some constraint.

Physical models of area-minimizing minimal surfaces can be made by dipping a wire frame into a soap solution, forming a soap ﬁlm, which is a minimal surface whose boundary is the wire frame. However no volume constraints are involved. We know that Clifford torus is a minimal surface in S3. The deﬁnition of the Clifford torus is given below.

Deﬁnition 2.29. In geometric topology, the Clifford torus is a torus in R4 deﬁned as

4 {(cos u, sin u, cos v, sin v) ∈ R | 0 < u < 2π, 0 < v < 2π} .

The so called Clifford torus in R3 is the stereographic projection of the actual Clifford torus in S3 and its parametrization is given by

R(u, v) = ((a + cos u) cos v, (a + cos u) sin v, sin u) (2.25) √ where a = 2.

The surfaces with constant H 6= 0 are called constant mean curvature (CMC) surfaces. The minimal surfaces are a subset of CMC surfaces. CMC surfaces play a prominent role in differential geometry. In 1841, Charles Eugene` Delaunay introduced a way of constructing rotationally symmetric CMC surfaces in R3, by prov- ing that a surface of revolution in R3 is a CMC surface if and only if its proﬁle curve

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is the roulette of a conic. A surface of revolution in R3 is generated by revolving a given proﬁle curve about a line in the plane containing this given curve [32, 33, 34]. The CMC surfaces have a lot of real world applications such as representations of soap bubbles, since they have the curvature corresponding to a nonzero pressure difference. Besides macroscopic bubble surfaces, CMC surfaces are relevant for other applications such as shape of the gasliquid interface on a superhydrophobic surface [9]. Now we deﬁne the Gaussian curvature or Gauss curvature of a surface.

Deﬁnition 2.30. The Gaussian curvature or Gauss curvature K of a surface at a

point is the product of the principal curvatures, k1 and k2, at the given point, i.e.

K = k1k2 (2.26)

It is an intrinsic measure of curvature, depending only on distances that are measured on the surface, not on the way it is isometrically embedded in any space. There are a lot of alternative deﬁnitions for the Gauss curvature in differential geometry. For example, it can be expressed as the ratio of the determinants of the second and ﬁrst fundamental forms:

det II ln − m2 K = = (2.27) det I EG − F 2 For an orthogonal parametrization (i.e., F = 0), the Gaussian curvature is given by 1 ∂ G 1 ∂ E 2 K = − √ √ u + √ u (2.28) 2 EG ∂u1 EG ∂u2 EG

Deﬁnition 2.31. If the induced metric is a (nonconstant) multiple of the ﬂat metric, then the immersion is said to be conformal, or isothermal, or in conformal

coordinates, i.e. ||ru1 || = ||ru2 || and hru1 , ru2 i = 0 at every point.

A surface is said to be (parameterized) in curvature line coordinates, if both ﬁrst and second fundamental forms are diagonal. A surface in R3 is said to be parameterized in isothermic coordinates if the given parameterization is conformal and in curvature line coordinates at the same time, i.e. it is given as an immersion

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2 3 2 2 r : D ⊂ R → R such that hru1 , ru2 i = 0, ||ru1 || = ||ru2 || = E, and for a smooth function over D, and hru1u2 , Ni = 0. Notice that obtaining a parameterization that is both isothermal and in curvature line coordinates at the same time is not possible on most regular surfaces [8]. The Gauss-Bonnet theorem is an important result in differential geometry which explains the connection between the geometry (in the sense of curvature) and the topology (in the sense of Euler characteristic).

Theorem 2.32. (Local version of Gauss-Bonnet theorem) Let U be a coordinate neighborhood of a two-dimensional Riemannian manifold. Let C be a simple sectionally smooth closed curve contained in U, positively oriented, parametrized by arc length s, and let

α1, α2, ...., αk be the internal angles at the vertices. Let R be the simply connected region bounded by C. Then

k Z Z X κg dS + K dS + (π − αi) = 2π (2.29) C R i=0 where κg is the geodesic curvature of C which measures the deviation of a curve from being a geodesic.

Theorem 2.33. (Global version of Gauss-Bonnet theorem) Let R ⊂ M be a regular region of a compact orientable two-dimensional Riemannian manifold. Let C1,C2, ...., Cn be closed, simple, sectionally differentiable curves forming the boundary ∂R. We assume that each Ci is positively oriented. Let αi, i = 1, ..., p, be the interior angles at the vertices of ∂R. Then we have

p Z Z X κg dS + K dS + (π − αi) = 2πχ(R) (2.30) ∂R R i=0

The following result is the famous ‘Theorema Egregium’- which characterizes the Gauss curvature as an intrinsic invariant of a surface.

Theorem 2.34. Let M1,M2 be two surfaces in Euclidean 3-space which are both diffeomorphic and isometric. Then the Gauss curvature K is the same at corresponding points [17].

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This means that the Gaussian curvature of a surface embedded in three-space is an intrinsic invariant to that surface even though K was deﬁned extrinsically.

2.4 Variations of a functional The functionals play a crucial role in analysis, mechanics, geometry and so on. The most advanced branch of the calculus of functionals is concerned with finding the extrema of the functionals which is called the calculus of variations. Actually, we will use a lot of techniques of calculus of variations in Willmore energy in the next chapters. Therefore, in this section, we briefly explain the concept of the first and second variations of a functional and some important theorems related to the variations of a functional. The reader can find the following definitions in [15].

Deﬁnition 2.35. Let us consider functional G[x] which is deﬁned on some normed linear space. Then G[x] is differentiable if its increment

∆G[x] = G[x + h] − G[x]

which can be written as ∆G[x] = µ[h] + ε||h||, (2.31)

where µ[h] is a linear functional and ε → 0 as ||h|| → 0. The quantity µ[h] is the principal linear part of the increment ∆G[x], and is called the ﬁrst variation (or ﬁrst differential) of G[x], denoted by δG[x].

Similarly, we deﬁne the second variation of G[x].

Deﬁnition 2.36. The functional G[x] is twice differentiable if its increment can be written in the form

2 ∆G[x] = µ1[h] + µ2[h] + ε||h|| , (2.32)

where µ1[h] is a linear functional (in fact, the ﬁrst variation), µ2[h] is a quadratic

functional, and ε → 0 as ||h|| → 0. This quadratic functional µ2[h] is called

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the second variation (or second differential) of the functional G[x], denoted by δ2G[h].

The following theorems can found in [15].

Theorem 2.37. A necessary condition for the differentiable functional G[x] to have an extremum for x =x ˆ is that its ﬁrst variation vanish for x =x ˆ, i.e., δG[h] = 0 for x =x ˆ and all admissible h.

The proof of this theorem can be found in [15]. Furthermore, the necessary condition can be explained using the second variation of the functional as follows:

Theorem 2.38. A necessary condition for the functional G[x] to have a minimum for x =x ˆ is that δ2G[x] ≥ 0 (2.33) for x =x ˆ and all admissible h. For a maximum, the sign ≥ in (2.33) is replaced by ≤.

See the proof in [15]. The above condition is not sufﬁcient to have a minimum of a functional. Now, the following theorem tells us the sufﬁcient condition for the functional to have a minimum for the given function.

Theorem 2.39. A sufﬁcient condition for a functional G[x] to have a minimum for x =x ˆ, given that the ﬁrst variation δG[h] vanishes for x =x ˆ, is that its second variation δ2G[h] be strongly positive for x =x ˆ.

Here, δ2G[h] which is deﬁned on some normed linear space is strongly positive if there exists a constant k > 0 such that δ2G[h] ≥ k||h||2 for all h.

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CHAPTER 3 WILLMORE ENERGIES

In this chapter we describe how the Willmore energy is deﬁned, and applied. The study of Willmore energy is an interesting subject which is connected to conformal geometry, geometric analysis, algebraic geometry, partial differential equations and geometric measure theory.

3.1 Deﬁnition of Willmore energy Willmore energies play a prominent role in differential geometry. In some sense, they capture the deviation of a surface from (local) sphericity. The ﬁrst in- vestigations on Willmore energies were performed by Poisson and Germain, inde- pendently from one another, at the beginning of the 19th century. The complete formalism is due to Willmore [17].

Deﬁnition 3.1. Let M be a smooth surface immersed in R3. We deﬁne the Will- more energy functional as Z W (M) = H2 dS, (3.1) M where the term dS is the area element with respect to the induced metric and H is the mean curvature.

In the literature, some authors [14] considered the Willmore functional as Z W˜ (M) = (H2 − K) dS (3.2) M

Sometimes, it is called classical bending energy and it can be easily shown that this functional has the same critical points as the original functional (3.1) using the Gauss-Bonnet formula. Therefore, the study of (3.1) is equivalent to the study of (3.2).

Theorem 3.2. Let M be a closed orientable surface and let f : M → R3 be a smooth

21 Texas Tech University, Thanuja Paragoda, May 2016 immersion of M into Euclidean 3-space. Then Z W (M) = H2 dS ≥ 4π. (3.3) M

Moreover, W (M) = 4π if and only if M is the round sphere embedded in R3 .

This theorem (in [17]) was easily proved by Willmore using the following in- equality Z K dS ≥ 4π, M where M is a compact immersed surface in R3 and Gauss curvature K ≥ 0. Fur- thermore, the equality holds if and only if k1 = k2 at every point. Further calcula- tions on a few examples of closed surfaces suggested that there could exist a better bound than (3.3) if the surface has genus g(M) > 0. Ultimately, Willmore proved the following theorem (see in [17]).

Theorem 3.3. Let M be a torus embedded in R3 as a tube of constant circular cross-section. Then Z H2 dS ≥ 2π2, (3.4) M the equality holding if and only if the generating curve is a circle and the ratio of the radii √ is 1/ 2.

In 1965, Willmore proposed to extend the previous result to any torus by formulating his famous conjecture:

“For every smooth torus M that is immersed in R3, W (M) ≥ 2π2 ”.

This was a challenge for mathematicians for over 45 years. In 2012, the conjecture was proved by Fernando Coda` and Andre` Arroja Neves using the min-max theory of minimal surfaces [11].

Theorem 3.4. Every embedded compact surface M in R3 with positive genus satisﬁes W (M) ≥ 2π2. Up to rigid motions, the equality holds only for stereographic projections of the Clifford torus.

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Willmore further characterized the Willmore energy as the invariant under conformal transformations in Euclidean space. He proved the following theorem using the fact that any conformal transformation of R3 can be decomposed into a product of similarity transformations and inversions.

Theorem 3.5. Let f : M → R3 be a smooth immersion of a compact orientable surface 3 into R . Let Z W (M) = H2 dS. M Then W (M) is invariant under conformal transformations of R3 [17].

This result was already known to Blaschke [38]. It is clear that this functional is invariant under similarity transformations and it was sufﬁcient to consider in- variance under inversions.

3.2 The Euler-Lagrange equation of the Willmore functional In this section we brieﬂy explain how we obtain the Euler-Lagrange equation of the Willmore functional which already proved in [35]. The history of this Euler- Lagrange equation is quite interesting. Originally, in 1929, it has been discovered in three-dimensional space by Schadow and also appeared in the PhD work of Gerhand Thomsen (see in [12]) who was a student of Wilhelm Blaschke. However, Konrad Voss also obtained this nice equation in the 1950s but he never published it. Then both Voss and Willmore were surprised that this precise equation was mentioned in Blaschke’s volume III in 1929. But its complete formalism was due to Willmore. Willmore also proved this Euler-Lagrange equation using the standard techniques of calculus of variations. He considered the normal variation of the immersion along with Gauss-Weingarten equations and Green’s second identity on closed surfaces [17]. Now we describe the notion of the smooth Willmore immersions. The following deﬁnition can be found in [35].

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Deﬁnition 3.6. Let s : M → Rn be a smooth immersion such that W (s) < ∞. s is a critical point for W if

∞ n d ∀w ∈ C (M, R ), W (s + tw) = 0. (3.5) dt t=0

Such an immersion is called Willmore. Then, the corresponding Euler-Lagrange equation which satisfies the above definition was obtained by Schadow and Thomsen. Theorem 3.7. A smooth immersion r : M → R3 is Willmore if and only if it solves the equation 2 ∆gH + 2H(H − K) = 0, (3.6) where ∆g is the Laplace Beltrami operator which depends on the natural metric. Now we further describe the minimizers of the Willmore functional. It is ob- vious that Willmore surfaces are generalizations of minimal surfaces. The stereographic projection of a minimal surface in S3 gives rise to a Willmore surface in R3 as proved by Lawson in 1970. In 2009, it was proved [30] that the existence of the flat minimizers of the Willmore functional restricted to a class of isometric immersions of the Riemannian surface (S, g) into R3 where S ⊂ R2 is a bounded C1 domain and g is the flat metric in R2. They derived the Euler-Lagrange equations satisfied by such constrained minimizers. The global minimizer of the Willmore energy in the class of tori is the Clifford torus [11]. Further, it was proved [7] that the homogenous tori are minimizers of their respective conformal classes near the Clifford torus. In 2014, M. Kilian, M.U. Schmidt and N. Schmitt proved a remarkable theorem (see theorem 3.8) which relates to the critical points of the Willmore energy. They studied equivariant “minimal tori and their stability under the Willmore energy and showed that all the spectral genus g = 1 minimal tori in S3 are unstable while the Clifford torus is stable” (see in [28]). Theorem 3.8. Amongst the equivariant CMC tori in S3, the Clifford torus is the only local minimum of the Willmore energy. All other equivariant minimal tori are local maxima of the Willmore energy [28].

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In our work, we study Generalized Willmore energy and ﬂow, which we will describe in the next chapters and provide a new numerical approach which is a novel numerical scheme on computing/constructing Willmore surfaces.

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CHAPTER 4 GENERALIZED WILLMORE ENERGIES

In this chapter, we introduce our Generalized Willmore energy on an immersed surface in R3 and study the time-dependent equation which describes the geometric evolution of a Generalized Willmore ﬂow. We consider both the general surface case and the graph case of the Generalized Willmore ﬂow equation.

4.1 Deﬁnition of Generalized Willmore energy Deﬁnition 4.1. The generalized Willmore energy functional associated to a surface M immersed in R3 is given by Z W (M) = (cH2 + µ) dS, (4.1) M where c = 2kc is the double of the usual bending rigidity kc and µ is the surface tension. The term dS is the area element with respect to the induced metric [2].

Then corresponding Euler-Lagrange equation of (4.1) is given by

2 ∆gH + 2H(H − K − ) = 0, (4.2) where = µ/c. Note that this represents a particular case of energies of the type Z (αH2 + βK + γ)dS M

(see for example [4]). The functional (4.1) has particular relevance for membranes with bending rigidity and surface tension, where other rigidities can be neglected [4]. The generalizations of the Willmore energy include the Helfrich energy for lipid bilayers, in which bending rigidities are established as multiplicative constants of the mean curvature and the Gauss curvature, while neglecting the surface tension of the membranes. In this work, our Generalized Willmore energy is a

26 Texas Tech University, Thanuja Paragoda, May 2016 combination of the classical bending energy (Willmore energy) and the energy due to the surface tension. Remark: Other mathematicians used the term Generalized Willmore energy for a more comprehensive type of energy, namely total free energy which includes curvature energy, volume energy and surface energy. It is expressed formally as I ε(H, K, t)dS, where t is the thickness of the focal conic domain; H and K are mean curvature and Gauss curvature of the inmost layer surface, respectively. Then, the corresponding Euler Lagrange equations to the free energy are as follows [40, 13]: I (∂ε/∂t)dS = 0 (4.3)

(∇2/2 + 2H2 − K)∂ε/∂H + (∇ · ∇˜ + 2KH)∂ε/∂K − 2Hε = 0 (4.4)

The deﬁnition of the operator ∇˜ can be found in the appendix of [39].

4.2 Generalized Willmore flow equation In this section we are interested in studying the Generalized Willmore flow, which is the L2-gradient flow corresponding to the Generalized Willmore energy. The Willmore flow usually presents in digital geometry processing, geometry modeling and physical simulation. In the literature, Droske and Rumpf derived a level set formulation for Willmore flow and they generalized the metric to sets of level set surfaces using the identification of normal velocities and variations of the level set function in time via the level set equation [27]. Ultimately, [14] studied the discrete Willmore energy and its flow and they derived the relevant gradient expres- sions including a linearization (approximation of the Hessian), which are required for nonlinear numerical solvers. In 2008, [37] presented a parametric approximation of Willmore flow and related geometric evolution equations. They provided various numerical simulations for energies appearing in the modeling of biological cell membranes. Some authors [20, 21] studied the error estimates for the Willmore flow of graphs along with numerical simulations.

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We are interested in formulating a PDE system for a Generalized Willmore ﬂow equation on the evolving surfaces and presenting a numerical approach using automatic differentation. Basically, we will consider two cases which are the general immersion and the graph case of evolving surfaces. Let us ﬁrst consider the evolution of a general surface M(t) which is embedded in R3. Let r : M ⊂ R2 → R3 be an immersed surface into three-dimensional Eu- clidean space which has the parametrization

r(u, v) = (X(u, v),Y (u, v),Z(u, v)) . (4.5)

Then we consider the following geometric evolution equation which is the Generalized Willmore ﬂow equation

2 V = ∆gH + 2H(H − K − ) (4.6)

on M(t), where t ∈ (0,T ) and V is the normal velocity of the evolving surfaces M(t) corresponding to the Generalized Willmore energy Z W (M) = (H2 + ) dS. M

(4.6) is a fourth-order ﬂow because the variation of the energy contains fourth- order derivatives. The surface M is evolving in time to follow variations of steepest descent of the energy. Now we reformulate (4.6) and obtain a PDE system for the evolving surfaces. We recall the ﬁrst fundamental form in matrix form: ! ! ! g11 g12 ru · ru ru · rv EF g = = = (4.7) g21 g22 rv · ru rv · rv FG Then normal vector N on the surface M is given by

ru × rv 1 N = = (YuZv − ZuYv,ZuXv − XuZv,XuYv − YuXv) (4.8) kru × rvk A where A = kru × rvk.

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The second fundamental form in matrix form: ! ruu · N ruv · N h = = ∇∇T r · N (4.9) rvu · N rvv · N where ! ∂u ∇(·) = . (4.10) ∂v

The Laplace-Beltrami operator in divergence form is given by

1 −1 ∆g(·) = ∇ · Ag ∇(·) (4.11) A

The mean curvature vector on M is

2HN = ∆r (4.12)

Using (4.11), (4.12) can be written as

1 2HN = ∇ · Ag−1∇r (4.13) A which is equivalent to

A 1 2HN = ∇ · g−1∇r + ∇A · g−1∇r (4.14) A A

Since ∇A · g−1∇r = 0, (4.14) becomes

2HN = ∇ · g−1∇r (4.15)

The Gauss curvature on M is

det(h) det(∇∇T r · N) K = = (4.16) det(g) A2

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where   ∂ur · ∂uN ∂ur · ∂vN T   ∇∇ r · N = −   (4.17) ∂vr · ∂uN ∂vr · ∂vN

Using (4.11) we can rewrite (4.6) as

1 ∂r ∇ · (Ag−1∇H) + 2H(H2 − K − ) = · N (4.18) A ∂t

Multiplying (4.18) by A and substituting (4.16) into (4.18), then (4.18) becomes

det(∇∇T r · N) ∂r ∇ · (Ag−1∇H) + 2HA H2 − − = A · N (4.19) A2 ∂t and multiplying (4.13) by A we obtain

2HAN = ∇ · Ag−1∇r (4.20)

Therefore, we transform (4.6) into a PDE system as follows:

det(∇∇T r · N) ∂r ∇ · (Ag−1∇H) + 2HA H2 − − = A · N (4.21) A2 ∂t 2HAN = ∇ · Ag−1∇r (4.22)

We solve this coupled PDE system using ﬁnite element methods along with the automatic differentiation which describes in the chapter 6.

4.3 Generalized Willmore equation on a graph In this section we describe the graph case of the Generalized Willmore flow. We first obtain the time dependent Generalized Willmore flow equation for the evolving surfaces. This is fourth order highly non linear PDE in the graph case. Then we will transform it into a system of second order non linear PDEs. We will

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use the description that we already published in [3]. Let M(t) = {(x, y, U(x, y, t)) | (x, y) ∈ Ω ⊂ R2, t ∈ [0,T ]} be the graph of U. Recall that the Laplace-Beltrami operator is given by the formula

1 ∇U(∇U)T ∆g = ∇ · AI − ∇ (4.23) A A

p 2 p 2 2 where A = 1 + |∇U| = 1 + Ux + Uy .

∆gH can be expressed as in the matrix form:

1 1 1 + U 2 −U U ∆ H = ∇ · y x y ∇H . g 2 (4.24) A A −UxUy 1 + Ux We can rewrite (4.24) as

1 ∇U(∇U)T 1 ∇U(∇U)T ∆gH = ∇· I− ∇(AH) −H∇· I− ∇A . (4.25) A A2 A A2

Using ∇U 2H = ∇ · (4.26) A we obtain

1 ∇U(∇U)T 1 ∆U ∇U I − ∇A = ∇A − ∇U + 2H . (4.27) A A2 A A A

It is straightforward to verify that

1 ∆U ∇ · ∇A − ∇U = −2K. (4.28) A A

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Using (4.27) and (4.28) we can express (4.24) as

1 ∇U(∇U)T ∇U ∆ H =∇ · I − (∇AH) + 2HK − 2H∇ · H g A A2 A 1 ∇U(∇U)T ∆gH =∇ · I − (∇AH) (4.29) A A2 ∇U + 2HK − ∇ · H2 − 2H3 . (4.30) A

By substituting (4.30) in (4.6) we obtain the fourth order PDE given by

1 ∇U(∇U)T ∇U V − ∇ · I − (∇AH) + ∇ · H2 + 2H = 0 . (4.31) A A2 A

Using (4.26) we rewrite (4.31) as

T 1 ∇U(∇U) 2 ∇U ∇U Ut − A∇ · I − (∇AH) − H − = 0 . (4.32) A A2 A A

where V = Ut/A. We express (4.6) in divergence form using the Gauss curvature formula of a graph as shown in (4.32). The important point is that the Gauss curvature does not appear in the fourth-order PDE. Then we transform the above fourth order PDE into a coupled system of two nonlinear second order PDEs as in [27, 19]. Deﬁne A ∇U W := ∇ · = AH (4.33) 2 A ! ∇U(∇U)T 1 1 + U 2 −U U B := I − = y x y . 2 2 2 2 (4.34) A 1 + Ux + Uy −UxUy 1 + Ux

We refer to W as weighted mean curvature. Then (4.32) becomes as a coupled system of two nonlinear second order PDEs with respect to two unknowns U and W ,

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B W 2 ∇U Ut = A∇ · ∇W − ∇U − (4.35) A A3 A A ∇U W = ∇ · . (4.36) 2 A

We solve this coupled PDE system numerically using ﬁnite element methods which describes in the chapter 6.

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CHAPTER 5 CLASSICAL AND DEFORMED WILLMORE ENERGIES IN SPACE FORMS

We study Willmore-type energies and Willmore-type surfaces in space forms and proved that Willmore-type immersions in different space forms essentially satisfy the same PDE (Willmore-type equation) in [5]. We ﬁrst brieﬂy introduce some notions related to space forms which can easily found in most Riemannian geometry text books. Then, we are interested in studying deformed Willmore energies in space form.

5.1 Sectional curvature and Space forms In Riemannian geometry, the sectional curvature is used to describe the curvature of Riemannian manifolds. It depends on a two-dimensional plane in the tangent space at a point and determines the curvature tensor completely. The more abstract deﬁnition of the sectional curvature is given below.

Definition 5.1. For any Riemannian manifold, and any linearly independent vector fields X,Y , the sectional curvature k(X,Y ) is defined as

hR(X,Y )Y,Xi k(X,Y ) = . hX,XihY,Y i − hX,Y i2 where R is the Riemannian curvature tensor which is given by

R(X,Y )Z = ∇X ∇Y Z − ∇Y ∇X Z − ∇[X,Y ]Z and X,Y,Z are arbitrary linearly independent vector ﬁelds, while the symbol ∇ denotes the usual Levi-Civita connection.

Notice that k(X,Y ) = hR(X,Y )Y,Xi ⇐⇒ X,Y are orthonormal. To avoid confusions, we will utilize the notation k(X,Y ) for a general sectional curvature, and k when the sectional curvature is a constant.

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Deﬁnition 5.2. A space form is a complete Riemannian manifold of constant sectional curvature k.

Examples: Euclidean n-space En, n-dimensional space Sn and hyperbolic space Hn. We are only interested in the 3-dimensional case. By rescaling the metric, there are three possible cases as follows.

Deﬁnition 5.3. Let M 3(k) be a 3-dimensional space form of constant sectional curvature k, namely,   3 4 S (k) = x ∈ R |hx, xi = 1/k if k > 0  3  M (k) = R3 if k = 0   3(k) = x ∈ 4|hx, xi = 1/k, x0 > 0 k < 0 H R H if where h·, ·i is the standard inner product on R4, while

1 1 2 2 3 3 0 0 hx, yiH = x y + x y + x y − x y

4 is the standard Lorenzian inner product on the Lorenz space R1. When k = 1, 0, −1, respectively, then M 3(k) is the standard unit sphere S3(1), the Euclidean space R3 or the hyperbolic space H3(−1) respectively.

They are the only connected, complete, simply connected Riemannian manifolds of given sectional curvature. All other connected complete constant curvature manifolds are quotients of those by some group of isometries.

5.2 Intrinsic versus extrinsic curvatures in space forms In Chapter 1, we explained some differential geometric theory of surface immersions in an ambient space such as curvature (mean, Gaussian, etc.) extrinsic and intrinsic curvatures. All types of curvature measure the “bending” of a surface in some sense (in one direction or another, along the principal directions, in

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arithmetic or geometric average, etc.). By deﬁnition, a curvature of an immersed surface is referred to as extrinsic if it takes into account the ambient space and its curvature. On the other hand, an intrinsic curvature is taking into account only the 2-dimensional manifold itself, regardless of the ambient space it is immersed in. No comprehensive and systematic study has been published so far regarding these notions for immersions into space forms. However, when it comes to a surface immersed in a space form of sectional curvature k, this “shift” from extrinsic to intrinsic mean curvatures is given by the formula

2 2 Hint = H + k, (5.1)

where H denotes the usual, extrinsic mean curvature, while Hint is the intrinsic mean curvature. Remark that (5.1) is strongly connected to that of Lawson’s 1-1 correspondence between families of isometric immersions with constant mean curvatures H,H0 into the space forms of sectional curvatures k, k0 respectively, as given by the relation:

2 H2 + k = H0 + k0. (5.2)

For more details on this relationship, one can consult [1] and [6]. More precisely, Lawson’s correspondence represents a particular case of the general philosophy of intrinsic versus extrinsic curvatures. Corresponding pairs of CMC surfaces surfaces in different space forms are informally called Lawson cousins. We remark that, in accordance with Lawson’s correspondence for CMC surfaces, the intrinsic mean curvature of a surface embedded in a space of sectional curvature k would be the same as the mean curvature of its Euclidean cousin. The natural question that one can pose is: could one formulate a 1-1 correspondence between (isometric families of) Willmore-type surfaces immersed in different space forms? The hidden reason beyond Lawson’s correspondence is a generalized harmonicity of the Gauss map of CMC surfaces. Hence, it becomes natural to consider the harmonicity of the Gauss map as the reason for a correspondence between Will-

36 Texas Tech University, Thanuja Paragoda, May 2016 more surfaces in space forms, as well.

5.3 Deformed Willmore energies in Euclidean space ˜ Deﬁnition 5.4. The deformed Willmore energy functional W (M; k1) for a surface M 3 immersed in the space form M (k0) with constant k1 is deﬁned as Z ˜ 2 W (M; k1) = (H + k1) dS. (5.3) M

˜ Theorem 5.5. Let us consider the deformed Willmore functional W (M; k1) of any smooth surface M immersed in M 3(0) = R3 Z ˜ 2 W = (H + k1) dS, (5.4) M where k1 is an arbitrary constant. Then, the Euler-Lagrange equation corresponding to the functional can be written as

2 ∆gH + 2H(H − K − k1) = 0. (5.5)

The previous theorem is not new. It was mentioned in several papers, including in [2], which shows a proof that is analogous to the original one, given by Willmore, for the case of the closed Willmore surfaces in Euclidean space - up to a change in terminology and notations.

Remark that the theorem simply states that: if one adds a constant term k1 to the integrand H2 of the Willmore energy of a surface M immersed in R3, then that will have a simple consequence to the Euler-Lagrange equation (Willmore equa- 2 tion), namely subtracting the term k1 from the quantity H − K.

This result motivated us to further investigate the following questions.

1. We are interested in deriving the Euler-Lagrange equation corresponding to the classical Willmore functional (3.1) if we consider surfaces immersed in a 3 general ambient space form M (k0) of constant sectional curvature k0.

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3 2. In a given ambient space form M (k0), we intend to determine the natural

choices of the constant k1 in the deformed Willmore functional (5.4) such that the corresponding Euler-Lagrange equation assumes the simpliﬁed form

2 ∆gH + 2H(H − K) = 0 , (5.6)

i.e., the same form that is obtained for the classical Willmore functional (3.1) in the Euclidean ambient space.

We show that the Euler-Lagrange equation takes this latter form only if we consider a new, extrinsic Laplace-Beltrami operator which would incorporate a term coming from the curvature k0 of the ambient space form. This represents our novel approach and interpretation, compared to the existing literature. Hence, we are able to identify different surfaces in different ambient spaces (the Euclidean space being one of them) which essentially share the same Euler-Lagrange PDE involving the mean and Gauss curvatures, eventually up to an extrinsic shift in the Laplace-Beltrami operator, as adapted to the speciﬁc ambient space.

5.4 Classical Willmore energies in space forms and their Euler-Lagrange equations The following theorem explains how to obtain the Euler-Lagrange equation from the classical Willmore energy in a space form M 3(k). This theorem can be found in [5], which was published in 2015. In this section, we are reproducing the results published in [5].

Theorem 5.6. Consider a given smooth immersion r of a surface M, with mean curva- 3 ture H and Gauss curvature K, in the space form M (k0) of sectional curvature k0. We consider two possible cases: Case 1): the surface M is closed and no boundary conditions have to be speciﬁed; ∂r Case 2): the surface M is not closed. In this case we assume that both r and ∂N are known smooth functions on the boundary ∂M. Let r be a minimizer of (3.1).

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Then, the mean curvature H of r must satisfy the equation

˜ 2 ∆gH + 2H(H − K + k0) = 0 , (5.7)

˜ (the Euler-Lagrange equation corresponding to the functional (3.1)), where ∆g represents 3 an extrinsic Laplace-Beltrami operator adapted to the space form M (k0).

Proof. Let us consider a smooth immersion r = r(u1, u2) of a surface M in the space 3 form M (k0) which represents a minimizer of (3.1) and such that either of the two cases of the hypothesis is satisﬁed (either Case 1, or Case 2). In order to determine the Euler-Lagrange equations, we consider the ﬁrst vari- 3 ation of (3.1) in M (k0). We have Z Z Z δ H2dS = 2HδH dS + H2δ(dS) . (5.8) M M M

In order to express the variation δH of the mean curvature, as well as the variation δ(dS) of the area element, some deﬁnitions in the Chapter 2 are used in this proof. We will use Einstein’s summation convention over repeated indices. We denote ∂r ∂N ∂2r ri = , Ni = , rij = (5.9) ∂ui ∂ui ∂ui∂uj j jk gij = hri, rji , hij = −hNi, rji , hi = g hki. (5.10)

j Here, the coefﬁcients hi represent the “contracted” second fundamental form (or shape operator). Using (5.10), the mean curvature and its ﬁrst variation are given by

1 ij H = g hij , (5.11) 2 1 ij 1 ij 1 ij δH = δ g hij = δ(g )hij + g δ(hij). (5.12) 2 2 2

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3 The Gauss and Weingarten equations in M (k0) are given by

k rij = Γijrk + hijN − gijk0r , (5.13) j Ni = −hi rj. (5.14)

k As usual, Γij denote the classical Christoffel symbols of the second kind. The normal variation of the immersion is deﬁned as

¯r(u1, u2, t) = r(u1, u2) + t φ(u1, u2) N , (5.15) where φ is a smooth real-valued function, and t is a real number such that − < t < for some > 0. Using (5.15) we have

δr = φN , δri = φiN + φNi , δrij = φijN + φiNj + φjNi + φNij . (5.16)

Using the deﬁnition in (5.15), it was proved in [2] that the variation of the area element is δ(dS) = −2φHdS. (5.17)

Let us turn to the variation δH. Again from (5.15), the ﬁrst variation of the ij normal vector is δN = −g φjri. Therefore, using (5.13) we have

pq k k hδN, riji = −hg φqrp, Γijrk + hijN − gijk0ri = −φkΓij + gijφk0 . (5.18)

This is due to the fact that hN, Ni = 1 and hN, Nii = 0. Moreover, we have (see [2], Theorem 5.10)

k hN, δriji = φij − φhi hjk. (5.19)

Then, using (5.18) we can express the variation of the hij tensor as

k k δhij = hδN, riji + hN, δriji = −φkΓij + gijφk0 + φij − φhi hjk = k = ∇i∇jφ − φhi hjk + gijφk0. (5.20)

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ij ik j Knowing that δ(g ) = 2φg hk (see e.g., [2] for details), we can ﬁnally obtain the desired variation of the mean curvature from (5.12) as

1 1 1 δH = 2φgikhj h + gij ∇ ∇ φ − φhkh + g φk = ∆ φ + φhi hk + 2k φ , 2 k ij 2 i j i jk ij 0 2 g k i 0 (5.21) where ∆g represents the intrinsic Laplace-Beltrami operator that depends only on 3 the metric of the surface, and does not depend on the ambient space form M (k0), namely: ij ij k ∆gφ = g ∇i∇jφ = g (φij − Γijφk). (5.22)

i k 2 2 Note that hkhi = trace(h ) = 4H − 2K + 2k0 and using (5.21) we have

2 2δH = ∆gφ + φ(4H − 2K + 4k0). (5.23)

Now let us consider Green’s Second Identity

Z I ∂φ ∂H (H∆gφ − φ∆gH) dS = H − φ dΓ, (5.24) M ∂M ∂N ∂N

∂φ where ∂N represents the directional derivative in the direction of the outward normal, considering that the boundary is traversed clockwise.

Case 1).: If the surface M is closed, then there is no boundary and the right- hand side of equation (5.24) is zero.

Case 2).: Otherwise, if the surface M is not closed, we assumed that both r and ∂r ∂N are known smooth functions on the boundary ∂M. In this case both the test ∂φ function φ and its normal derivative ∂N vanish on the boundary, therefore also the right-hand side of equation (5.24) vanishes. Therefore, one has Z Z H∆gφ dS = φ∆gH dS. (5.25) M M

Combining the variations (5.17), (5.23), and using (5.25), the ﬁrst variation of the

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Willmore functional is Z 2 δW = φ(∆gH + 2H(H − K + 2k0)) dS. (5.26) M

Therefore the Euler-Lagrange equation corresponding to the functional is

2 ∆gH + 2H(H − K + 2k0) = 0. (5.27)

If instead of the intrinsic Laplace-Beltrami operator we use by deﬁnition the ˜ extrinsic Laplace-Beltrami operator ∆g acting on any smooth function ψ of the given coordinates:

˜ ij ij k ∆gψ = g ∇i∇jψ + 2k0ψ = g (ψij − Γijψk) + 2k0ψ, (5.28)

then the Willmore equation can be rewritten in the equivalent form

˜ 2 ∆gH + 2H(H − K + k0) = 0 , (5.29)

The proof is now complete.

Remark 5.7. Other choices of boundary conditions can also be considered involv- ∂H ing H and ∂N , but in this case extra boundary integrals should be included in functional (3.1). Remark 5.8. The motivation we considered when introducing the notion of “extrinsic operator” is primarily for the convenience of writing the Willmore equation in a uniform way, regardless of the ambient space form in which the surface is immersed. On the other hand, there exists a striking similarity between the Laplace-Beltrami operator that we deﬁned extrinsically, and the well-known Hodge-Laplace operator acting on 1-forms. Existing literature introduced several types of Laplace operators. As a reminder, they are deﬁned as:

• The Hodge Laplacian is a differential operator is a differential operator acting

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on differential forms (on any Riemannian manifold),

∆ = dδ + δd = (d + δ)2,

where d is the exterior derivative and δ is the coderivative.

• The connection Laplacian, is the same as the Laplace-Beltrami operator, is deﬁned on any tensor, as 2 ∆gT = tr∇ T,

where 2 ∇X,Y T = ∇X ∇Y T − ∇∇X Y T.

2 If the connection is the Levi-Civita connection, then ∇ = ∆g can be written as the Laplace-Beltrami operator 1 p ij ∆gf = p ∂i |detg|g ∂jf |detg|

where f is a scalar function. The Hodge-Laplace operator ∆ and the Laplace- Beltrami operator, as deﬁned on differential forms in general, are related via the Weitzenbock-Bochner formula

∆ = ∇2 + Ric,

where Ric is an appropriately deﬁned Ricci type tensor. It is very important to note that for 0-forms (that is, functions), Ric ≡ 0, and hence, the two Laplacians coin- cide. On the other hand, if one would apply the Weitzenbock-Bochner¨ formula to a 1-form like α = f(x1, x2)(dx1 + dx2) + dx3, where {x1, x2} are the normal coordinates on the surface immersed in the space form M 3, and x3 is normal to the surface, then the connection operator ∇2, as restricted to the surface, represents the Laplace-Beltrami operator ∆g, while Ric(α), where α is the given form, actu-

ally becomes the operator 2k0 · id, where k0 is the sectional curvature of the space ∂ 1 2 3 form, namely Ric(α, X) ≡ 2k0 · f, when X = ∂x3 , and α = f(dx + dx ) + dx . This means that our extrinsic Laplace-Beltrami operator is essentially the same as the

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Hodge-Laplace operator applied to the 1-form α.

5.5 Comparative Analysis A). We would like to point out that the Willmore equations published in both [18] and [22] were only considered and proved for closed surfaces, but not for surfaces with a boundary, while we considered both of these cases. We are grateful to both works, as they represented the foundational, ground-breaking resources for this comparative study, in which we claim that they are in complete agreement. B). With the remark A). in place regarding closed surfaces, the Willmore equation (5.27) that we obtained is equivalent to the equation (3.18) obtained in [22] when using the intrinsic Laplace-Beltrami operator. On the other hand, for consis- tency with the classical Willmore PDE, we are using the extrinsic Laplace operator. C). We would like to point out that the authors of [22] thought that their equation (3.18) contradicted the equation (3.13) from [18]. More speciﬁcally, they stated: ”If we try to apply the ﬁnal equation (3.13) [18] for Willmore surfaces in S3, we obtain instead of the second term in brackets in (3.18) the wrong expression . . . ”. However, we found that the two equations mentioned in these two different references are equivalent. The generalized Willmore equation (3.13) published in [18] was written as (case m = 2): 3 0 0 a b ∆gH + 2H + H(R − R − R abN N ) = 0, (5.30)

0 0 where R ij is the Ricci tensor of the ambient space, and R is the scalar curvature corresponding to the Ricci tensor of the ambient space. As always, H and K represent the mean and Gauss curvature of the immersed surface respectively. It is straightforward that this equation is equivalent with the Willmore equation (5.27) which was ﬁrst stated by [22] for the closed surfaces in S3, and generalized by us. So, we found that there exists no inconsistency between [22] and [18] and, aside from the notational setting, the Willmore-type equations are fully equivalent. We consider both of these references as fully rigorous and mathematically cor- rect and are in agreement with both. We are just seeking a novel interpretation, context and viewpoint.

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5.6 Deformed Willmore energies in space forms and their Euler-Lagrange equations Theorem 5.9. (reproduced from our paper [5]) Consider a surface M that is immersed in 3 ˜ the ambient space M (k0). The deformed Willmore energy functional W (M; k1) has the corresponding Euler-Lagrange equation

˜ 2 ∆gH + 2H(H − K − k1 + k0) = 0 , (5.31)

˜ 3 where ∆g is the extrinsic Laplace-Beltrami operator of the immersion in M (k0).

Proof. Based on the previous two theorems 5.5 and 5.4 presented in this work, the proof is immediate.

In particular, when k1 = k0 in the previous theorem, we obtain an Euler- Lagrange equation that has the same form as that for surfaces embedded in R3. ˜ Corollary 5.10. The Euler-Lagrange equation for W (M; k0), where M is immersed in the 3 space form M (k0), is ˜ 2 ∆gH + 2H(H − K) = 0. (5.32)

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CHAPTER 6 NUMERICAL RESULTS

In this chapter we solve several cases for the PDE systems we computed in Chapter 4 corresponding to the Generalized Willmore flow based on finite element methods. Our PDE systems are linearized in practice with automatic differentiation tools and we provide non-trivial solutions, including a Clifford torus for the PDE systems subject to both Dirichlet and Neumann boundary conditions. Some convergence results and some graphical figures are presented for the Generalized Willmore flow of graphs.

6.1 Automatic Differentiation (AD) for exact Jacobian computation of the PDE systems In computational mathematics, automatic differentiation, sometimes called al- gorithmic differentiation or computational differentiation is a set of techniques to numerically evaluate the derivative of a function specified by a computer program. It utilizes the fact that every computer program, no matter how complicated, exe- cutes a sequence of elementary arithmetic operations (addition, subtraction, mul- tiplication, division, etc.) and elementary functions (exp, log, sin, cos, etc.). By applying the chain rule repeatedly to these operations, derivatives of arbitrary order can be computed automatically accurately to working precision. AD is neither a symbolic differentiation nor a numerical differentiation. The classical methods such as symbolic differentiation and numerical differentiation produce some is- sues in calculating higher order derivatives. For instance, the symbolic differentiation heads to inefficient code (unless carefully done) and features the difficulty of converting a computer program into a single expression, while numerical differentiation is subjected to round-off errors in the discretization process and significant digit cancellation [23]. Therefore, these classical methods are performing either at a low speed or approximation at computing the partial derivatives of a function with respect to many inputs. AD solves all of these problems at the expense of introducing more software dependencies. In our work we have chosen to use the Adept software library, which enables

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algorithms written in C++ to be automatically differentiated using an operator overloading strategy. The full Jacobian matrix can be computed with very little code modiﬁcation [31]. Differentiation can be performed in forward mode, reverse mode, or the full Jacobian matrix can be computed. We use the functionality of computing the full Jacobian matrix of the FEMuS library which is a FINITE ELE- MENT MULTIGRID SOLVER.

6.2 Time dependent solutions of the Generalized Willmore equation on a general surface: Dirichlet boundary conditions We explain how we obtain the time dependent weak formulation for (4.19) and (4.20). 1 Multiplying (4.19) by test function φ0 ∈ H0 (Ω) and then integrating over the domain Ω, we obtain Z Z −1 2 ∂r ∇ · (Ag ∇H) + 2HA(H − K − ) φ0 dΩ = A · N φ0 dΩ (6.1) Ω Ω ∂t

1 Multiplying (4.20) by the test function φ1∈ H0(Ω) and integrating over Ω, we obtain Z Z −1 Ag 2HAN · φ1 dΩ = ∇ · Ag ∇r · φ1 dΩ, (6.2) Ω Ω

Then we apply integration by parts for the ﬁrst part of (6.1) and right hand side of (6.2) and obtain the following system

Z Z −1 −1 − Ag ∇H · ∇φ0 dΩ + Ag ∇H · νφ0 dΓ Ω Γ Z Z 2 ∂r + 2HA(H − K − )φ0 dΩ = A · N φ0 dΩ, (6.3) Ω Ω ∂t

Z Z Z −1 −1 2HAN · φ1 dΩ = − Ag ∇r : ∇φ1 + Ag ∇r · N · φ1 dΓ (6.4) Ω Ω Γ

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where Γ is the boundary of Ω. In the case of Dirichlet boundary conditions the boundary terms vanish in the above PDE system because of the choice of the test

function φ0 and φ1. Therefore, the weak formulation of the time-dependent Generalized Willmore ﬂow on a general surface is given by

Z −1 − Ag ∇H · ∇φ0 dΩ Ω Z Z 2 ∂r 1 + 2HA(H − K − )φ0 dΩ = A · N φ0 dΩ, ∀φ0 ∈ H0 (Ω) (6.5) Ω Ω ∂t

Z Z −1 1 2HAN · φ1 dΩ = − Ag ∇r : ∇φ1 dΩ ∀φ1 ∈ H0(Ω) (6.6) Ω Ω

with prescribed Dirichlet boundary conditions H(Γ) = H0 and r(Γ) = r0. The above nonlinear coupled PDE system with respect to four unknowns X,Y,Z and H is solved by using a Newton scheme. We make use of AD tools to evaluate the jacobian of this coupled PDE system. Let a = (X,Y,Z,H) and we rewrite (6.5) and (6.6) in compact notation as

F(a) = 0 .

Let a0 be an initial guess. Then a single Newton iteration is given by

−1 an = an−1 − J (an−1)F(an−1) for n ≥ 1 (6.7)

∂F and it is repeated until ||an − an−1|| < . Here, J(an) = ∂a (an).

6.3 Computational models of the Clifford torus In order to validate our numerical algorithm, we have compared steady-state Willmore solutions with known analytical Clifford torus solutions. Using (2.25) we have

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X = (a + cos u) cos v (6.8) Y = (a + cos u) sin v (6.9) Z = sin u (6.10) 1 cos u H = + (6.11) 2 2(a + cos u)

which also provide the Dirichlet boundary conditions for X,Y,Z and H respectively at u = 0, 2π, v = 0, 2π. Since, topologically a rectangle is the fundamental polygon of a torus with opposite edges sewn together, we consider the square domain as the polygon of the Clifford torus.

Figure 6.1. The domain in the xy-plane of the Clifford torus

The ﬁgures 6.2,6.3, 6.4 and 6.5 represent the proﬁles of X,Y,Z and the mean curvature H respectively.

6.4 Steady-state graph solutions of the Willmore equation: Dirichlet boundary conditions 6.4.1 Finite element weak formulation We describe how we obtain the steady state weak formulation for the above coupled PDE system. From now on we set Ut = 0 in (4.35). Now we multiply

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Figure 6.2. The X-proﬁle of the Clifford torus with = 0

1 (4.35) and (4.36) by test functions ϕ, ψ ∈ H0 (Ω) respectively and apply integration by parts we obtain the following system

Z B Z B Z W 2 − ∇W · ∇ϕ dΩ + ∇W · νϕ dΓ + 3 ∇U · ∇ϕ dΩ (6.12) Ω A Γ A Ω A Z W 2 Z ∇U Z ∇U − 3 ∇U · νϕ dΓ + · ∇ϕ dΩ − · νϕ dΓ = 0 (6.13) Γ A Ω A Γ A Z ∇U Z ∇u Z W − ∇ψ dΩ + · νψ dΓ = 2 ψ dΩ (6.14) Ω A Γ A Ω A where Γ is the boundary of Ω. In the case of Dirichlet boundary conditions the boundary terms vanish in the above formulation because of the choice of the test functions ϕ and ψ. Hence, the weak formulation of the steady-state Generalized Willmore ﬂow

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Figure 6.3. The Y -proﬁle of the Clifford torus with = 0 graph is given by

Z 2 B W 1 − ∇W · ∇ϕ + 3 + ∇U · ∇ϕ dΩ = 0 , ∀ϕ ∈ H0 (Ω) (6.15) Ω A A A Z 2W ∇U 1 ψ + · ∇ψ dΩ = 0 , ∀ψ ∈ H0 (Ω) (6.16) Ω A A with prescribed Dirichlet boundary conditions U(Γ) = U0 and W (Γ) = W0. Let v = (U, W ) and we rewrite (6.15) and (6.16) in compact notation as

F(v) = 0 .

Let v0 be an initial guess. Then a single Newton iteration is given by

−1 vn = vn−1 − J (vn−1)F(vn−1) for n ≥ 1 (6.17)

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Figure 6.4. The Z-proﬁle of the Clifford torus with = 0

∂F and it is repeated until ||vn − vn−1|| < . Here, J(vn) = ∂v (vn).

6.4.2 The sphere as a Steady-State Solution of the Willmore ﬂow In order to validate our numerical algorithm, we compare steady-state Will- more ﬂow solutions with known analytical solutions. We test not only that each solution converges to the corresponding analytic one, but also that the theoretical convergence order is recollected for h → 0, where h is the size of the mesh domain discretization.

We consider a sector of a sphere whose projection is the unit circle on the xy- plane. p Let z = U(x, y) = R2 − x2 − y2 for x2 + y2 ≤ 1, where R = sec θ for π/2 ≤ θ ≤ 5π/2.

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Figure 6.5. The mean curvature of the Clifford torus with = 0

We compute −x Ux = , (6.18) pR2 − x2 − y2 and

−y Uy = , (6.19) pR2 − x2 − y2

p 2 p 2 2 Since A = 1 + |∇U| = 1 + Ux + Uy , using (6.18) and (6.19) we obtain

R A = , (6.20) pR2 − x2 − y2 Using (6.18), (6.19) and (6.20) we compute

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∇U x y = − i − j, (6.21) A R R Then we obtain

∇U 2 ∇ · = − , (6.22) A R By substituting (6.22) and (6.20) into (4.33) we obtain 1 W = − , (6.23) pR2 − x2 − y2 Then the Dirichlet boundary conditions for U and W are given by

1 √ U θ= π = U θ= 5π = , (6.24) 6 6 3 √ W π = W 5π = − 3. (6.25) θ= 6 θ= 6

where θ is the angle measured from positive x-axis to the spherical cap.

Table 6.1. L2 error and order of convergence for spherical cap Linear Quadratic Bi-quadratic Level Error Order Error Order Error Order 1 1.4679e-01 6.4171e-03 5.5830e-03 1.763 2.843 2.892 2 4.3246e-02 8.942e-04 7.5201e-04 1.924 3.192 3.210 3 1.1397e-02 9.7854e-05 8.1292e-05 1.979 3.221 3.177 4 2.8918e-03 1.0491e-05 8.9854e-06 1.994 3.146 3.073 5 7.2574e-04 1.1851e-06 1.0679e-06

Tables 6.1 and 6.2 show the error in the L2 norm, the error in the seminorm, and the corresponding convergence order, for different types of ﬁnite element families: piecewise linear, piecewise quadratic and piecewise bi-quadratic. All errors vanish as h → 0. According to the convergence results this is well performing and

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Table 6.2. Seminorm error and order of convergence for spherical cap Linear Quadratic Bi-quadratic Level Error Order Error Order Error Order 1 4.3248e-01 8.1004e-02 6.9570e-02 0.756 1.656 1.642 2 2.5608e-01 2.5700e-02 2.2291e-02 0.914 1.934 1.916 3 1.3594e-01 6.7248e-03 5.9078e-03 0.975 2.042 2.000 4 6.9168e-02 1.6324e-03 1.4768e-03 0.993 2.059 2.009 5 3.474e-02 3.9176e-04 3.6690e-04 the theoretical asymptotic errors are recovered for each ﬁnite element family: 2 for linear, 3 for quadratic and 3 for bi-quadratic in the L2 norm error 1 for linear, 2 for quadratic and 2 for bi-quadratic in the semi norm error respectively.

Figure 6.6. The sector of a spherical cap proﬁle U with = 0

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Figure 6.7. The sector of spherical cap weighted mean curvature W with = 0

6.4.3 The Clifford torus as a Steady-State Solution of the Willmore flow In this section, we compare the numerical solution of the Willmore graph with a sector of the Clifford torus. Other comparisons that have been done show similar results. Using the coefficients of the first and second fundamental forms it can be easily shown that the profile, the mean curvature, the area element and the weighted mean curvature of the Clifford torus in R3 are given respectively by

U = sin u, (6.26) 1 cos u H = + , (6.27) 2 2(a + cos u) 1 A = , (6.28) sin u 1 1 cos u W = AH = + . (6.29) sin u 2 2(a + cos u)

For u = 0 and u = π (the limits in u for the half-torus graph) the area element degenerates, A → ∞, and for this reason we chose only the sector

π 5π ≤ u ≤ and 0 ≤ v ≤ 2π. 6 6

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The domain in the xy-plane is then given by the annulus with the following inner and outer radii 5π √ √ rin = a + cos = 2 − 3/2, (6.30) 6 π √ √ rout = a + cos = 2 + 3/2. (6.31) 6

The Dirichlet boundary conditions for U and W are also given by

1 U u= π = U u= 5π = , (6.32) 6 √6 √2 √ √ 2 + 3 2 − 3 √ √ √ √ W u= π = 2 and W u= 5π = 2 . (6.33) 6 2 2 + 3 6 2 2 − 3

We numerically solve (6.15) and (6.16) with the help of the FEMuS library, an open- source Finite Element Multiphysics Solver which uses automatic differentiation to evaluate the exact Jacobian in the Newton iteration scheme.

Table 6.3. L2 error and order of convergence for Clifford torus Linear Quadratic Bi-quadratic Level Error Order Error Order Error Order 1 8.0250e-01 3.1005e-01 2.9624e-01 0.816 0.478 0.431 2 4.5592e-01 2.2258e-01 2.1978e-01 0.603 1.592 1.612 3 3.0010e-01 7.3818e-02 7.1887e-02 1.055 3.702 3.692 4 1.4447e-01 5.6711e-03 5.5611e-03 2.278 3.694 3.687 5 2.9795e-02 4.3813e-04 4.3169e-04

Tables 6.3 and 6.4 show the error in the L2 norm, the error in the seminorm, and the corresponding convergence order, for different types of ﬁnite element families: piecewise linear, piecewise quadratic and piecewise bi-quadratic. All errors vanish as h → 0. Concerning the error in the L2 norm, the theoretical asymptotic convergence orders are recovered for each ﬁnite element family: 2 for linear, 3 for quadratic and 3 for bi-quadratic. Also for the error in the seminorm the theoretical

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Table 6.4. Seminorm error and order of convergence for Clifford torus Linear Quadratic Bi-quadratic Level Error Order Error Order Error Order 1 1.4669 7.2594e-01 6.9164e-01 0.448 0.529 0.480 2 1.0754 5.0318e-01 4.9600e-01 0.523 1.514 1.529 3 7.4860e-01 1.7614e-01 1.7187e-01 0.717 2.635 2.619 4 4.5557e-01 2.8358e-02 2.7980e-02 1.420 2.023 2.016 5 1.7031e-01 6.9777e-03 6.9158e-03 asymptotic convergence orders are recovered for each ﬁnite element family: 1 for linear, 2 for quadratic and 2 for bi-quadratic.

6.4.4 Generalized Willmore Torus In this section we solve the Generalized Willmore equation system (6.15)– (6.16) on the same domain and with the same boundary conditions for U and W as in the previous section, but we set the parameter 6= 0. The resulting graph is not a Clifford torus anymore but a deformation of it, and an analytic solution is no longer available for comparison. From now on, we refer to each numerical solution obtained with 6= 0 as Generalized Willmore torus. In Fig. 6.8, we show the variation of the proﬁle cross section for the General- ized Willmore torus with three different values of , namely = 0.001, 0.01 and 0.03. Note that by increasing the value of the cross section moves farther away from the one of the Clifford torus (which corresponds to = 0). Finally, in Figures 6.9 and 6.10 we show the color maps of the proﬁle U and the generalized curvature W for the Generalized Willmore torus with = 0.01.

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1.2

A B U 0.5 1

π/6 C 0

0.8

0.5 1 1.5 2 2.5 1 1.2 1.4 x x

Figure 6.8. On the left: cross section of some computed Generalized Willmore torus surfaces for the angle range π/6 ≤ u ≤ 5π/6, for various values of . The dashed curve is the Clifford torus. The curves for = 0.03 (A), = 0.01 (B) and = 0.001 (C) are visible on the right, in a zoom of the dash-dotted rectangle.

Figure 6.9. Generalized Willmore torus proﬁle U with = 0.01

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Figure 6.10. Generalized Willmore torus weighted curvature W with = 0.01

6.4.5 The Catenoid as a Steady-State Solution of the Willmore flow We compare steady-state Willmore flow graph solutions with known analytical catenoidal solutions subject to the Dirichlet boundary conditions. In the steady case, every minimal surface , i.e. every surface with H ≡ 0, solves the coupled system for the graph case which describes in the chapter 4 and hence is a Will- more surface. In fact, minimal surfaces are the steady state trivial solutions of the Willmore flow. As [24] states, its authors provided “two sharp sufficient conditions for immersed Willmore surfaces in R3 to be already minimal surfaces which means to have vanishing mean curvature on their entire domains and these results are appli- cable for Willmore graphs.” Solving the boundary value problem for the Willmore equation leads us to the catenoid. More precisely, the authors show that Willmore graphs on bounded C4 domains with vanishing mean curvature on the boundary must be already minimal graphs. For example, imposing H = 0 on the two cir- cles of an annulus, we will only obtain the catenoid as a solution of the Willmore equation. The catenoid can be described by following parametric equations. v x = c cosh cos u, (6.34) c v y = c cosh sin u, (6.35) c z = v, (6.36)

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where u ∈ [0, 2π), v ∈ R and c is a non-zero real constant. It can be easily shown that the proﬁle and the area element which are respectively given by

1p U = v = c cosh−1 x2 + y2 , (6.37) c 1 A = . (6.38) v

Since the mean curvature of the catenoid is zero, the weighted mean curvature is also zero. i.e. W = AH = 0. The domain in the xy-plane is given by the annulus with the following radii which is the same domain we considered for the Clifford torus.

√ √ rin = 2 − 3/2, √ √ rout = 2 + 3/2.

The Dirichlet boundary conditions for U and W are also given by

−1 rin −1 rout U = c cosh and U = c cosh , (6.39) c c rin rout

W = W = 0. (6.40) rin rout

We numerically solve (6.15) and (6.16) (in the case of = 0) with help of AD tools in the Newton scheme. We consider the case of c = 0.5 for the catenoid. The tables (6.5) and (6.6) show us the error in the L2 norm, the error in the semi norm and the corresponding convergence order for the catenoid for different types of ﬁnite element families: piecewise linear, piecewise quadratic and piecewise bi- quadratic. According to the convergence results it is under performing and the theoretical asymptotic convergence orders are gradually recovered for each ﬁnite element family: 1 for linear, 2 for quadratic and 2 for bi-quadratic. If c > 0.54,

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Table 6.5. L2 error and order of convergence for Catenoid Linear Quadratic Bi-quadratic Level Error Order Error Order Error Order 1 1.6084e-01 5.8436e-02 5.4530e-02 1.113 1.614 1.544 2 7.4342e-02 1.9093e-02 1.8702e-02 1.358 1.942 1.926 3 2.8996e-02 4.9677e-03 4.9221e-03 1.579 2.375 2.368 4 9.7038e-03 9.5791e-04 9.5339e-04 1.773 2.774 2.771 5 2.8393e-03 1.4003e-04 1.3972e-04

Table 6.6. Seminorm error and order of convergence for Catenoid Linear Quadratic Bi-quadratic Level Error Order Error Order Error Order 1 9.8467e-01 3.2705e-01 3.1349e-01 0.739 0.732 0.687 2 5.9015e-01 1.9687e-01 1.9474e-01 0.717 0.970 0.960 3 3.5902e-01 1.0053e-01 1.0012e-01 0.793 1.283 1.280 4 2.0727e-01 4.1309e-02 4.1227e-02 0.885 1.584 1.583 5 1.1227e-01 1.3774e-02 1.3758e-02 the solution diverges. From 0 < c < 0.53, the solutions converges well. It is numerically proved that if M is a graph over domain in plane such that H ≡ 0 on the boundary, then H ≡ 0 everywhere (see the ﬁgure (6.12) ).

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Figure 6.11. Catenoid proﬁle U

6.5 Steady-state graph solutions of the Willmore equation: Dirichlet-Neumann boundary conditions We explain how we obtain the steady state weak formulation for the coupled PDE system in the graph case subject to the Dirichlet-Neumann boundary conditions.

6.5.1 Finite element weak formulation By applying the same procedure followed in the case of the Dirichlet boundary 1 conditions problem we multiply (4.35) and (4.36) by test functions ψ0 ∈ H0 (Ω) and 1 ψ1 ∈ H (Ω) respectively and apply integration by parts we obtain the following

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Figure 6.12. Weighted mean curvature W of the catenoid system

Z B W 2 − ∇W · ∇ψ0 + 3 + ∇U · ∇ψ0 dΩ Ω A A A Z 2 B W ∇U 1 + ∇W − 3 ∇U − · νψ0 dΓ = 0, ∀ψ0 ∈ H0 (Ω) (6.41) Γ A A A Z W ∇U 1 2 ψ1 − · ∇ψ1 dΩ = 0 ∀ψ1 ∈ H (Ω) (6.42) Ω A A with prescribed Dirichlet-Neumann boundary conditions

∂U(Γ) ∇U U(Γ) = U0 and = · nˆ, ∂n A

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where Γ is the boundary of Ω. In the case of Dirichlet-Neumann boundary conditions the boundary terms do not vanish in the above formulation.

6.5.2 The Clifford torus as a Steady-State Solution of the Willmore ﬂow with Dirichlet-Neumann boundary conditions Let us consider x = (a + cos(u)) cos(v) , y = (a + cos(u)) sin(v) and z = sin(v). p 2 1 Then z = ±(1 − (a − x2 + y2) ) 2 . Using (6.26) and (6.28) one can easily obtain that ∇U a − px2 + y2 a − px2 + y2 = xi + yj. (6.43) A px2 + y2 px2 + y2

For the inner boundary of the torus, the normal vector (denoted by n1) is given by

xi + yj xi + yj xi + yj n1 = = = √ √ . p 2 2 x + y rin 2 − 3/2

Using (6.43) we compute √ √ ∇U 3/2 3/2 = √ √ xi + √ √ yj (6.44) A 2 − 3/2 2 − 3/2 on the inner boundary. Then √ ∇U 3 · n = . A 1 2 Similarly, we take

−xi − yj −xi − yj −xi − yj n2 = = = √ √ . p 2 2 x + y rout 2 + 3/2

on the outer boundary. Then using (6.43) we compute √ √ ∇U 3/2 3/2 = − √ √ xi + − √ √ yj (6.45) A 2 + 3/2 2 + 3/2

on the outer boundary.

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Then √ ∇U 3 · n = . A 2 2 The Dirichlet-Neumann boundary conditions for U and W are given by

1 U = U = √ rin rout 3 √ ∂U ∂U 3 = = . ∂n ∂n 2 1 rin 2 rout respectively.

Table 6.7. L2 error and order of convergence for Clifford torus with Dirichlet- Neumann boundary conditions Linear Quadratic Bi-quadratic Level Error Order Error Order Error Order 1 8.1102e-01 2.1752e-01 1.4070e-01 1.243 2.186 1.798 2 3.4272e-01 4.7814e-02 4.0452e-02 1.546 3.074 2.543 3 1.1735e-01 5.6799e-03 6.9423e-03 1.777 2.327 3.121 4 3.4236e-02 1.1318e-03 7.9816e-04 1.917 1.672 3.364 5 9.0658e-03 3.5510e-04 7.7528e-05

The tables (6.7) and (6.8) show us the error in the L2 norm, the error in the seminorm and the corresponding convergence order for the Clifford torus for different types of finite element families: piecewise linear, piecewise quadratic and piecewise bi-quadratic. According to the convergence results, in the fifth mesh level, the errors are close to zero in the quadratic and bi-quadratic finite element families and therefore the finite element numerical solutions converge to the analytical Clifford torus solutions in the case Dirichlet-Neumann boundary conditions. In Figures 6.13 and 6.14 we show the color maps of the profile U and the weighted mean curvature W of the Clifford torus under Dirichlet-Neumann boundary conditions. The color maps 6.15 and 6.16 present the Dirichlet-Neumann

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Table 6.8. Seminorm error and order of convergence for Clifford torus with Dirichlet-Neumann boundary conditions Linear Quadratic Bi-quadratic Level Error Order Error Order Error Order 1 1.4694 6.0123e-01 5.0045e-01 0.580 1.225 1.033 2 9.8298e-01 2.5717e-01 2.4457e-01 0.775 1.528 1.492 3 5.7434e-01 8.9168e-02 8.6937e-02 0.913 1.762 1.758 4 3.0497e-01 2.6289e-02 2.5706e-02 0.974 1.912 1.905 5 1.5526e-01 6.9861e-03 6.8644e-03 boundary conditions of U and W respectively.

Figure 6.13. Clifford torus proﬁle U subject to the Dirichlet-Neumann boundary conditions

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Figure 6.14. Weighted mean curvature W subject to the Dirichlet-Neumann boundary conditions

Figure 6.15. Dirichlet-Neumann boundary conditions for U

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Figure 6.16. Dirichlet-Neumann boundary conditions for W

6.6 The disk of genus of two as a steady-state graph solution of the Willmore flow In this section, we investigate higher genus numerical steady-state graph solutions of the Willmore flow. Therefore, we change the domain which we used in the case of the Clifford torus and the catenoid. We consider the disk with radius 3 with two holes whose radii are 0.5 (see the figure 6.17). Then we solve our coupled PDE system in the graph case using AD tools over the domain of disk with two holes. In the non-linear iterations, in order to get the stability of the solutions we use as initial guess the solution of two Laplace problem with the same boundary condition for U and W . Therefore, in the first iteration, we solve the usual Laplace equation and then in the rest of the non linear iterations we solve our coupled PDE system subject to the Dirichlet boundary conditions.. In fact, we do not have the comparison results because the analytical solutions are unknown in this case. But the figures 6.18 and 6.19 show up the profile of the required graph and the corresponding weighted mean curvature. In fact, if the (weighted) mean curvature vanishes on the boundaries, then it is zero everywhere by [24], in the sense that the Willmore graph becomes a minimal graph.

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Figure 6.17. Disk with two holes

The Dirichlet boundary conditions for U and W are given by

U = 1.5 and U = 0 R=3 r=0.5

W = W = 0 R=3 r=0.5 .

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Figure 6.18. Proﬁle of a disk with two holes

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Figure 6.19. The weighted mean curvature

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CHAPTER 7 CONCLUSIONS

In this dissertation a novel numerical scheme for solving a Generalized Will- more flow equation has been presented. We studied both case of general immersion and the graph case of Generalized Willmore flow. In both cases, we formu- lated the geometric evolution equations as coupled systems of nonlinear PDEs where the unknowns are the profile and the weighted mean curvature. To solve these coupled systems we made use of AD techniques to compute the Jacobian in the Newton linearization of the finite element weak formulation. We have tested the accuracy of the algorithm by providing nontrivial steady-state numerical solutions of the Generalized Willmore flow equation. The present work can be ex- tended in several directions. The numerical scheme and the implementation presented here can be applied to time-dependent problems and to conformal immersions in R3. These studies are expected to bring new interesting results which will be the subject of future works. In addition, we also studied Willmore-type energies and Willmore-type immersions in space forms. We have deduced the Euler-Lagrange equation of the deformed Willmore energy in a space form, in a unified way, using an extrinsic Laplace-Beltrami operator. In fact, we considered both the case of closed surfaces and the one of surfaces with boundary, for which we gave and discussed the necessary boundary value conditions, which the previous literature failed to do. This is a contemporary approach in the study of Willmore energies in space forms. It has been proved simultaneously by several mathematicians that the harmonicity of the conformal Gauss map represents an equivalent condition for an immersed surface to be Willmore, in S3. For different space forms, the generalized Gauss map is defined differently; nevertheless, a specific type of harmonicity property will be in place. We will show in a future study that the correspondence between Willmore surfaces immersed in different space forms is based on the harmonicity of certain generalized Gauss maps.

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BIBLIOGRAPHY

[1] Fujioka A. Harmonic maps and associated maps from simply connected Rie- mann surfaces into the 3-dimensional space forms. Tohoku Mathematical Jour- nal, Second Series, 47(3):431–439, 1995. 36

[2] Athukoralage B. Capillarity and Elastic Membrane Theory from an Energy Point of View. PhD thesis, Texas Tech University, Aug 2014. 26, 37, 40, 41

[3] Athukoralage B., Aulisa E., Bornia G., Paragoda T., and Toda M. New ad- vances in the study of Generalized Willmore surfaces and ﬂow. Seventeenth International Conference on Geometry, Integrabiity and Quantization, pages 1–11, 2015. 31

[4] Athukoralage B. and Toda M. Geometric models for secondary structures in proteins. AIP Proceedings - American Institute of Physics, 1558(883), 2013. 26

[5] Athukoralage B., Paragoda T., Toda M., and Bornia G. Willmore- type energies and Willmore-type surfaces in space forms. JP Journal of Geometry and Topology, 18:93–108, 2015. 34, 38, 45

[6] Lawson H. B. Complete minimal surfaces in S3. Annals of Mathematics, pages 335–374, 1970. 36

[7] Ndiaye C.B. and Schatzle¨ R.M. Explicit conformally constrained Willmore minimizers in arbitrary codimension. Calc. Var., 51:291–314, 2014. 24

[8] Aulisa E., Toda M., and Kose Z. Constructing isothermal curvature line coordinates on surfaces which admit them. Central European Journal of Mathematics - Versita-Springer Verlag, 11:1982–1993, 2013. 18

[9] Lobaton E.J. and Salamon T.R. Computation of constant mean curvature surfaces: Application to the gasliquid interface of a pressurized ﬂuid on a superhydrophobic surface. Journal of Colloid and Interface Science., 314(1):184–198, 2007. 17

[10] Harley F. Differential forms with applications to the physical sciences. Dover, 1989. 11

[11] Marques F. and Neves A. Min-max theory and the Willmore conjecture. An- nals of Mathematics, 179(2):683–782, 2014. 22, 24

74 Texas Tech University, Thanuja Paragoda, May 2016

[12] Thomsen G. Uber` konforme geometrie 1: Grundlagen der konformen ﬂachentheorie.` Lutcke & Wulff, 1923. 23

[13] Naito H., Okuda M., and Ou-Zang Z.C. Preferred shapes of smetic-a phase grown from isotropic phase: Origin of Focal Conic Domains. Phys.Rev.E, 52:2095–2098, 1995. 27

[14] Bobenko A. I. and Schroder P. Discrete Willmore ﬂow. Eurographics Symposium on Geometry Processing, pages 101–110, 2005. 21, 27

[15] Gelfand I.M. and Fomin S.V. Calculus of Variations. Dover, 1963. 19, 20

[16]J urgen` J. Riemannian Geometry and Geometric Analysis. Berlin: Springer-Verlag, 2002. 11

[17] Willmore T. J. Riemannian Geometry. Oxford Science Publications. Oxford University Press, 1993. 3, 4, 15, 18, 21, 22, 23

[18] Willmore T. J. and Jhaveri C. An extension of a result of Bang-Yen Chen. The Quarterly Journal of Mathematics, 23(1):319–323, 1972. 44

[19] Deckelnick K., Dziuk G., and Elliott C.M. Computation of geometric partial differential equations and mean curvature ﬂow. Acta Numerica, Cambridge University Press, 14:139–232, 2005. 32

[20] Deckelnick K., Katz J., and Schieweck F. A C1-ﬁnite element method for the Willmore ﬂow of two dimensional graphs. Mathematics of computation, 84(296):2617–2643, 2015. 27

[21] Ji L. and Xu Y. Optimal error estimates of the local discontinuous Galerkin method for Willmore ﬂow of graphs on cartesian meshes. International Journal of Numerical Analysis and Modeling, 8(2):252–283, 2011. 27

[22] Kholodenko A. L. and Nesterenko V. V. Classical dynamics of the rigid string from the Willmore functional. JINR preprint E2-93-181, 1993. 44

[23] Bartholomew-Biggs M., Brown S., Christianson B., and Dixon L. Automatic differentiation of algorithms. Computational and Applied Mathematics, 124 (1- 2):171–190, 2000. 46

[24] Bergner M. and Jacob R. Sufﬁent conditions for Willmore immersions R3 to be minimal surfaces. Ann Glob Anal Geom, 45:129–146, 2014. 2, 60, 69

[25] Do Carmo M. Differential forms and applications. Universitext. Springer Berlin Heidelberg, 1994. 3, 4, 6

75 Texas Tech University, Thanuja Paragoda, May 2016

[26] Do Carmo M. Riemannian Geometry, Mathematics: Theory & Applications. Birkhauser` Boston, 2013. 3, 4, 6, 7, 8, 9, 10

[27] Droske M. and Rumpf M. A level set formulation for Willmore ﬂow. Interfaces free boundaries, pages 361–378, 2004. 27, 32

[28] Kilian M., Schmidt M.U., and Schmitt N. On stability of equivariant minimal tori in the 3-sphere. Geometry and Physics, 85:171–176, 2014. 24

[29] Obata M. Conformal transformations of Riemannian manifolds. Differential geometry, 4:311–333, 1970. 8

[30] Hornung P. Flat minimizers of the Willmore functional:Euler-Lagrange equations. Communications on Pure and Applied Mathematics, 64(3):367–441, 2011. 24

[31] Hogan R.J. Adept fast automatic differentiation library for C++: User guide. Document version 1.0, 2013. 47

[32] Nahid S. Explicit parametrization of Delaunay surfaces in space forms via loop group methods. Kobe Journal of Mathematics, 22:71–107, 2005. 17

[33] Paragoda T. Constant mean curvature surfaces of revolution versus Willmore surfaces of revolution: A comparative study with physical applications. Mas- ter’s thesis, Texas Tech University, May 2014. 17

[34] Paragoda T., Athukoralage B., and Toda M. Roulettes of conics, Delaunay surfaces and applications. Surveys in Mathematics, 4:1–23, 2014. 17

[35] Riviere` T. Weak immersions of surfaces with l2-bounded second fundamental form. PCMI Graduate Summer School, 2013. 23

[36] Willmore T.J. An introduction to differential geometry. Oxford University Press, 1959. 3, 5

[37] Barrett J. W., Garcke H., and Nurberg R. Parametric approximation of Will- more ﬂow and related geometric evolution equations. SIAM J. Sci. Comput., 31(1):225–253, 2008. 27

[38] Blaschke W. Einfuhrung in die Differentialgeometrie. 1950. 23

[39] Tu Z.C. and Ou-Yang Z.C. Geometry theory on the elasticity of bio- membranes. Journal of Physics A: Mathematical and General, 37:11407–11429, 2004. 27

76 Texas Tech University, Thanuja Paragoda, May 2016

[40] Tu Z.C. and Ou-Yang Z.C. Variational problems in elastic theory of biomem- branes, smectic-a liquid crystals, and carbon related structures. Seventh Inter- national Conference on Geometry, Inegrability and Quantization, pages 237–248, 2005. 27

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CHAPTER 8 APPENDIX: COMPUTER CODE

// The following compute code solves Generalized Willmore flow equation in the graph case

#include”FemusInit.hpp” #include”MultiLevelProblem.hpp” #include”NumericVector.hpp” #include”VTKWriter.hpp” #include”GMVWriter.hpp” #include”NonLinearImplicitSystem.hpp” #include”adept.h” #include

using namespace femus;

i n t simulation = 2; // =1 sphere(default)=2 torus //Sphere double thetaSphere = acos( −1.) / 3 ; bool SetBoundaryConditionSphere(const std::vector < double >& x , const char SolName[] , double& value , const int facename, const double time) { bool dirichlet = true; //dirichlet i f (!strcmp(”u”, SolName)) { value = tan(thetaSphere); } e l s e if (!strcmp(”W”, SolName)) { value = −1. / tan(thetaSphere); }

return dirichlet; } double InitalValueUSphere(const std::vector < double >& x ) { return tan(thetaSphere); } double InitalValueWSphere(const std::vector < double >& x ) { return −1. / tan(thetaSphere); }

// Torus bool SetBoundaryConditionTorus(const std::vector < double >& x , const char SolName[] , double& value , const int facename, const double time) { bool dirichlet = true; //dirichlet

double theta = acos( −1.) / 6 ; double z = sin(theta);

i f (!strcmp(”u”, SolName)) { value = z ; } e l s e if (!strcmp(”W”, SolName)) {

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double theta1 = theta; double theta2 = acos( −1.) − theta1 ; doubleA= 1. /z; double H1 = 0.5 ∗ (1. + cos(theta1) / (sqrt(2.) + cos(theta1))); double H2 = 0.5 ∗ (1. + cos(theta2) / (sqrt(2.) + cos(theta2)));

i f (facename == 1) { value = − A ∗ H1 ; } e l s e if (facename == 2) { value = − A ∗ H2 ; } } return dirichlet; } double InitalValueUTorus(const std::vector < double >& x ) { double r = sqrt(x[0] ∗ x [ 0 ] + x [ 1 ] ∗ x [ 1 ] ) ; double cosu = r − s q r t ( 2 ) ; return sqrt(1 − cosu ∗ cosu ) ; //return 0.5; }

double InitalValueWTorus(const std::vector < double >& x ) { double r = sqrt(x[0] ∗ x [ 0 ] + x [ 1 ] ∗ x [ 1 ] ) ; double cosu = r − s q r t ( 2 ) ; double sinu = sqrt(1 − cosu ∗ cosu ) ; return − 0 . 5 / sinu ∗ (1 + cosu / (sqrt(2.) + cosu)); //return 0.25; }

void AssembleWillmoreProblem AD(MultiLevelProblem& ml prob ) ; std : : pair < double, double > GetErrorNorm( MultiLevelSolution ∗ mlSol ) ; i n t main(int argc, char ∗∗ args ) { i f (argc >= 2) { i f (!strcmp(”sphere”, args[1]) | | ! strcmp (”Sphere”, args[1]) | | ! strcmp (”SPHERE”, args[1])) { simulation = 1;

i f (argc >= 3) { std::string str; std::stringstream ss; ss << args [ 2 ] ; ss >> s t r ; i n t angle = atoi(str.c s t r ( ) ) ; thetaSphere = acos( −1.) / 180 ∗ angle ; } } e l s e if (!strcmp(”torus”, args[1]) | | ! strcmp (”Torus”, args[1]) | | ! strcmp (”TORUS”, args[1])) simulation = 2; e l s e { std : : cout << ”Wrong input, using d e f a u l t argument: simulation

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= 1 (Sphere)” << std : : endl ; } } e l s e { std : : cout << ”No input argument, using d e f a u l t argument: simulation = 1 (Sphere)” << std : : endl ; }

FemusInit mpinit(argc , args , MPI COMM WORLD) ; unsigned maxNumberOfMeshes; maxNumberOfMeshes = 5; vector < vector < double > > l2Norm; l2Norm. resize (maxNumberOfMeshes); vector < vector < double > > semiNorm; semiNorm. resize (maxNumberOfMeshes); f o r(unsignedi=0;i < maxNumberOfMeshes; i ++) { std :: ostringstream filename; i f (simulation == 1) { filename << ”./input/circle q u a d” << i << ”.neu”; } e l s e if (simulation == 2) { filename << ”./input/torus30 ” << i << ”.neu”; }

MultiLevelMesh mlMsh; double scalingFactor = 1.; //mlMsh. ReadCoarseMesh(”./input/circle q u a d.neu”,”seventh”, scalingFactor); mlMsh.ReadCoarseMesh(filename . str (). c s t r ( ) ,”seventh”, scalingFactor); unsigned dim = mlMsh.GetDimension();

unsigned numberOfUniformLevels = 1; unsigned numberOfSelectiveLevels = 0; mlMsh. RefineMesh(numberOfUniformLevels , numberOfUniformLevels +numberOfSelectiveLevels , NULL); mlMsh. EraseCoarseLevels (numberOfUniformLevels − 1 ) ; mlMsh. PrintInfo ();

FEOrder feOrder[3] = {FIRST , SERENDIPITY, SECOND} ; l2Norm[i ]. resize (3); semiNorm[i ]. resize (3);

f o r(unsignedj=0; j < 3 ; j ++) { // loop on theFE Order MultiLevelSolution mlSol(&mlMsh);

mlSol .AddSolution(”u” , LAGRANGE, feOrder [ j ] ) ; mlSol .AddSolution(”W” , LAGRANGE, feOrder [ j ] ) ;

i f (simulation == 1) { mlSol. Initialize(”u”, InitalValueUSphere); mlSol. Initialize(”W”, InitalValueWSphere); mlSol . AttachSetBoundaryConditionFunction(SetBoundaryConditionSphere ); mlSol . GenerateBdc(”u”); mlSol . GenerateBdc(”W”);

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} e l s e if (simulation == 2) { mlSol. Initialize(”u”, InitalValueUTorus); mlSol. Initialize(”W”, InitalValueWTorus); mlSol . AttachSetBoundaryConditionFunction(SetBoundaryConditionTorus ); mlSol . GenerateBdc(”u”); mlSol . GenerateBdc(”W”); }

NonLinearImplicitSystem& system = mlProb.add system < NonLinearImplicitSystem > (”Willmore”);

system . AddSolutionToSystemPDE(”u”); system . AddSolutionToSystemPDE(”W”);

system . SetAssembleFunction(AssembleWillmoreProblem AD); system. init (); system.MGsolve();

std : : pair< double, double > norm = GetErrorNorm(&mlSol ); l2Norm[i][j] = norm. first ; semiNorm[i ][ j ] = norm.second;

std :: vector < std:: string > variablesToBePrinted ; variablesToBePrinted .push back (”All”);

VTKWriter vtkIO(&mlSol ); vtkIO. SetSurfaceVariable(”u”); vtkIO . write (DEFAULT OUTPUTDIR,”biquadratic”, variablesToBePrinted , i); } } std : : cout << std : : endl ; std : : cout << ”l2 ERROR and ORDER OF CONVERGENCE: \ n\n”; std : : cout << ”LEVEL\tFIRST\ t \ t \tSERENDIPITY\ t \tSECOND\n”; f o r(unsignedi=0;i < maxNumberOfMeshes; i ++) { std : : cout << i + 1 << ”\ t”; std ::cout.precision(14);

f o r(unsignedj=0; j < 3 ; j ++) { std : : cout << l2Norm[i ][ j ] << ”\ t”; } std : : cout << std : : endl ;

i f (i < maxNumberOfMeshes − 1) { std ::cout.precision(3); std : : cout << ”\ t \ t”;

f o r(unsignedj=0; j < 3 ; j ++) { std : : cout << log(l2Norm[i][j] / l2Norm[i + 1][j]) / log(2.) << ”\ t \ t \ t”;

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} std : : cout << std : : endl ; } } std : : cout << std : : endl ; std : : cout << std : : endl ; std : : cout << ”SEMINORM ERROR and ORDER OF CONVERGENCE: \ n\n”; std : : cout << ”LEVEL\tFIRST\ t \ t \tSERENDIPITY\ t \tSECOND\n”;

f o r(unsignedi=0;i < maxNumberOfMeshes; i ++) { std : : cout << i + 1 << ”\ t”; std ::cout.precision(14);

f o r(unsignedj=0; j < 3 ; j ++) { std : : cout << semiNorm[ i ][ j ] << ”\ t”; }

std : : cout << std : : endl ;

i f (i < maxNumberOfMeshes − 1) { std ::cout.precision(3); std : : cout << ”\ t \ t”;

f o r(unsignedj=0; j < 3 ; j ++) { std : : cout << log(semiNorm[i][j] / semiNorm[i + 1][j]) / log(2.) << ”\ t \ t \ t”; } std : : cout << std : : endl ; } }

return 0; } /∗∗ ∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗ / void AssembleWillmoreProblem AD(MultiLevelProblem& ml prob ) {

adept::Stack& s = FemusInit:: adeptStack ;

NonLinearImplicitSystem ∗ mlPdeSys = &ml prob . get system (”Willmore”); const unsigned level = mlPdeSys−>GetLevelToAssemble (); const unsigned levelMax = mlPdeSys−>GetLevelMax (); const bool assembleMatrix = mlPdeSys−>GetAssembleMatrix ();

Mesh∗ msh = ml prob . ml msh−>GetLevel(level ); elem∗ e l = msh−>e l ;

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MultiLevelSolution ∗ mlSol = ml prob . m l s o l ; Solution ∗ s o l = ml prob . ml sol−>GetSolutionLevel(level );

LinearEquationSolver ∗ pdeSys = mlPdeSys−> LinSolver[level ]; SparseMatrix ∗ KK = pdeSys−> KK ; NumericVector∗ RES = pdeSys−> RES; const unsigned dim = msh−>GetDimension (); unsigned iproc = msh −>p ro ce ss or i d ( ) ; unsigned soluIndex; soluIndex = mlSol−>GetIndex (”u”); unsigned soluType = mlSol−>GetSolutionType(soluIndex ); unsigned soluPdeIndex; soluPdeIndex = mlPdeSys−>GetSolPdeIndex(”u”); vector < adept :: adouble > solu ; unsigned solWIndex; solWIndex = mlSol−>GetIndex (”W”); unsigned solWType = mlSol−>GetSolutionType(solWIndex ); unsigned solWPdeIndex; solWPdeIndex = mlPdeSys−>GetSolPdeIndex(”W”); vector < adept :: adouble > solW ; vector < vector < double > > x ( dim ) ; unsigned xType = 2; vector< i n t > KKDof ; vector phi ; vector phi x ; vector phi xx ; vector< double > Res ; vector< adept :: adouble > aResu ; vector< adept :: adouble > aResW ; const unsigned maxSize = static c a s t < unsigned >(ceil(pow(3, dim)));

solu. reserve(maxSize); solW. reserve(maxSize); f o r(unsignedi=0;i < dim ; i ++) x[i ]. reserve(maxSize); sysDof. reserve(2 ∗ maxSize ) ;

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phi. reserve(maxSize); phi x. reserve(maxSize ∗ dim ) ; unsigned dim2 = (3 ∗ ( dim − 1) + ! ( dim − 1 ) ) ; phi xx. reserve(maxSize ∗ dim2 ) ;

Res. reserve(2 ∗ maxSize ) ; aResu. reserve(maxSize); aResW. reserve(maxSize ); vector < double > J a c ; Jac.reserve(4 ∗ maxSize ∗ maxSize ) ; vector< double > J a c t ; Jact .reserve(4 ∗ maxSize ∗ maxSize ) ;

KK−>zero ( ) ;

f o r(int iel = msh −>IS Mts2Gmt elem offset[iproc ]; i e l < msh−>IS Mts2Gmt elem offset[iproc + 1]; iel++) { unsigned kel = msh−>IS Mts2Gmt elem [ i e l ] ; unsigned kelGeom = el−>GetElementType(kel );

unsigned nDofs = el−>GetElementDofNumber(kel , soluType ); unsigned nDofs2 = el−>GetElementDofNumber(kel , xType); KKDof. resize (2 ∗ nDofs ) ; solu. resize(nDofs); solW. resize(nDofs); f o r(inti=0;i < dim ; i ++) { x[i ]. resize(nDofs2); }

Res. resize(2 ∗ nDofs ) ; aResu. resize(nDofs); aResW. resize (nDofs);

std:: fill(aResu.begin(), aResu.end(), 0); std:: fill (aResW.begin() , aResW.end() , 0);

f o r(unsignedi=0;i < nDofs ; i ++) { unsigned iNode = el−>GetMeshDof(kel , i , soluType); unsigned solDof = msh−>GetMetisDof(iNode, soluType ); solu [ i ] = (∗ sol−> Sol[soluIndex])(solDof); solW [ i ] = (∗ sol−> Sol[solWIndex])( solDof ); KKDof[i] = pdeSys−>GetKKDof(soluIndex , soluPdeIndex , iNode); KKDof[nDofs + i] = pdeSys−>GetKKDof(solWIndex , solWPdeIndex , iNode); }

f o r(unsignedi=0;i < nDofs2; i++) { unsigned iNode = el−>GetMeshDof(kel , i , xType);

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unsigned xDof = msh−>GetMetisDof(iNode, xType);

f o r(unsigned idim = 0; idim < dim; idim++) { x[idim][i] = (∗msh−> coordinate−> Sol[idim])(xDof); } }

s . new recording (); // ∗∗∗ Gauss point loop ∗∗∗ f o r(unsigned ig = 0; ig < msh−> finiteElement [kelGeom][soluType]−>GetGaussPointNumber ( ) ; ig ++) { msh−> finiteElement [kelGeom][soluType]−>Jacobian(x, ig, weight, phi, phi x , phi xx ) ;

adept::adouble soluGauss = 0; vector < adept :: adouble > soluGauss x ( dim , 0 . ) ;

adept::adouble solWGauss = 0; vector < adept :: adouble > solWGauss x ( dim , 0 . ) ;

vector < double > xGauss(dim, 0.);

f o r(unsignedi=0;i < nDofs ; i ++) { soluGauss += phi[i] ∗ solu [ i ] ; solWGauss += phi[i] ∗ solW [ i ] ; f o r(unsigned idim = 0; idim < dim; idim++) { soluGauss x[idim] += phi x [ i ∗ dim + idim ] ∗ solu [ i ] ; solWGauss x[idim] += phi x [ i ∗ dim + idim ] ∗ solW [ i ] ; xGauss[idim] += x[idim][ i] ∗ phi [ i ] ; } }

double c= 0.; double Id[2][2] = {{ 1 . , 0 . } , { 0 . , 1 . }} ; adept::adouble A2 = 1.; vector < vector < adept :: adouble> > B( dim ) ;

f o r(unsigned idim = 0; idim < dim; idim++) { B[idim]. resize(dim); A2 += soluGauss x [ idim ] ∗ soluGauss x [ idim ] ; }

adept::adouble A = sqrt(A2);

f o r(unsigned idim = 0; idim < dim; idim++) { f o r(unsigned jdim = 0; jdim < dim; jdim++) { B[idim][jdim] = Id[idim][jdim] − ( soluGauss x [ idim ] ∗ soluGauss x[jdim]) / A2; } }

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// ∗∗∗ p h i i loop ∗∗∗ f o r(unsignedi=0;i < nDofs ; i ++) {

adept::adouble nonLinearLaplaceU = 0.; adept :: adouble nonLinearLaplaceW = 0.; f o r(unsigned idim = 0; idim < dim; idim++) {

nonLinearLaplaceU += − 1 . / A ∗ soluGauss x [ idim ] ∗ phi x [ i ∗ dim + idim ] ; nonLinearLaplaceW += −1. / A ∗ ((B[idim][0] ∗ solWGauss x [ 0 ] +B[idim][1] ∗ solWGauss x [ 1 ] ) − ( solWGauss ∗ solWGauss / A2 + c) ∗ soluGauss x [ idim ] ) ∗ phi x [ i ∗ dim + idim ] ; }

aResu[i] += (2. ∗ solWGauss / A ∗ phi [ i ] − nonLinearLaplaceU) ∗ weight ; aResW[ i ] += nonLinearLaplaceW ∗ weight ; } // end phi i loop } // end gauss point loop

f o r(inti=0;i < nDofs ; i ++) { Res[i] = aResu[i].value(); Res[nDofs + i] = aResW[i].value(); }

RES−>add vector blocked(Res, KKDof);

s.dependent(&aResu[0] , nDofs); s.dependent(&aResW[0] , nDofs);

s.independent(&solu[0] , nDofs); s.independent(&solW[0] , nDofs);

s.jacobian(&Jac [0]);

f o r(int inode = 0; inode < 2 ∗ nDofs; inode++) { f o r(int jnode = 0; jnode < 2 ∗ nDofs; jnode++) { J a c t [ inode ∗ 2 ∗ nDofs + jnode] = −J a c [ jnode ∗ 2 ∗ nDofs + inode]; } }

KK−>add matrix blocked(Jact , KKDof, KKDof); s . clear independents (); s . clear dependents (); } } RES−>c l o s e ( ) ; KK−>c l o s e ( ) ; }

// functions post processing double GetExactSolutionValueSphere(const std::vector < double >& x ) {

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return sqrt(1. / (cos(thetaSphere) ∗ cos(thetaSphere)) − x [ 0 ] ∗ x [ 0 ] − x [ 1 ] ∗ x [ 1 ] ) ; } ;

void GetExactSolutionGradientSphere(const std::vector < double >& x , vector < double >& solGrad ) { double pi = acos( − 1 . ) ; solGrad[0] = −x[0] / sqrt(1. / (cos(thetaSphere) ∗ cos(thetaSphere)) − x [ 0 ] ∗ x [ 0 ] − x [ 1 ] ∗ x [ 1 ] ) ; solGrad[1] = −x[1] / sqrt(1. / (cos(thetaSphere) ∗ cos(thetaSphere)) − x [ 0 ] ∗ x [ 0 ] − x [ 1 ] ∗ x [ 1 ] ) ; } ;

double GetExactSolutionValueTorus(const std::vector < double >& x ) { double r = sqrt(x[0] ∗ x [ 0 ] + x [ 1 ] ∗ x [ 1 ] ) ; double cosu = r − s q r t ( 2 ) ; return sqrt(1 − cosu ∗ cosu ) ; } ;

void GetExactSolutionGradientTorus(const std::vector < double >& x , vector < double >& solGrad ) { double r = sqrt(x[0] ∗ x [ 0 ] + x [ 1 ] ∗ x [ 1 ] ) ; double cosu = r − s q r t ( 2 ) ; double z = sqrt(1 − cosu ∗ cosu ) ; solGrad[0] = − cosu / ( r ∗ z ) ∗ x[0]; solGrad[1] = − cosu / ( r ∗ z ) ∗ x [ 1 ] ; } ;

std : : pair < double, double > GetErrorNorm( MultiLevelSolution ∗ mlSol ){ unsigned level = mlSol−> ml msh−>GetNumberOfLevels () − 1u ; Mesh∗ msh = mlSol−> ml msh−>GetLevel(level ); elem∗ s o l = msh−>e l ; Solution ∗ s o l = mlSol−>GetSolutionLevel(level );

const unsigned dim = msh−>GetDimension (); unsigned iproc = msh−>pr o ce ss or id ( ) ;

unsigned soluIndex; soluIndex = mlSol−>GetIndex (”u”); unsigned soluType = mlSol−>GetSolutionType(soluIndex );

vector < double > solu ;

vector < vector < double > > x ( dim ) ; unsigned xType = 2;

vector phi ; vector phi x ;

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vector phi xx ;

const unsigned maxSize = static c a s t < unsigned >(ceil(pow(3, dim))); solu. reserve(maxSize);

f o r(unsignedi=0;i < dim ; i ++) x[i ]. reserve(maxSize);

phi. reserve(maxSize); phi x. reserve(maxSize ∗ dim ) ; unsigned dim2 = (3 ∗ ( dim − 1) + ! ( dim − 1 ) ) ; phi xx. reserve(maxSize ∗ dim2 ) ;

double seminorm = 0.; double l2norm = 0.;

f o r(int iel = msh −>IS Mts2Gmt elem offset[iproc ]; i e l < msh−>IS Mts2Gmt elem offset[iproc + 1]; iel++) { unsigned kel = msh−>IS Mts2Gmt elem [ i e l ] ; unsigned nDofs = el−>GetElementDofNumber(kel , soluType ); unsigned nDofs2 = el−>GetElementDofNumber(kel , xType);

solu. resize(nDofs);

f o r(inti=0;i < dim ; i ++) { x[i ]. resize(nDofs2); }

f o r(unsignedi=0;i < nDofs2; i++) { unsigned iNode = el−>GetMeshDof(kel , i , xType); unsigned xDof = msh−>GetMetisDof(iNode, xType);

f o r(unsigned idim = 0; idim < dim; idim++) { x[idim][i] = (∗msh−> coordinate−> Sol[idim])(xDof); } }

// ∗∗∗ Gauss point loop ∗∗∗ f o r(unsigned ig = 0; ig < msh−> finiteElement [kelGeom][soluType]−>GetGaussPointNumber ( ) ; ig+ +) { msh−> finiteElement [kelGeom][soluType]−> Jacobian(x, ig, weight, phi, phi x , phi xx ) ;

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double soluGauss = 0; vector < double > soluGauss x ( dim , 0 . ) ; vector < double > xGauss(dim, 0.);

f o r(unsignedi=0;i < nDofs ; i ++) { soluGauss += phi[i] ∗ solu [ i ] ;

f o r(unsigned idim = 0; idim < dim; idim++) { soluGauss x[idim] += phi x [ i ∗ dim + idim ] ∗ solu [ i ] ; xGauss[idim] += x[idim][ i] ∗ phi [ i ] ; } }

double exactSol; vector solGrad(dim);

i f (simulation == 1) { exactSol = GetExactSolutionValueSphere(xGauss); GetExactSolutionGradientSphere(xGauss, solGrad ); } e l s e if (simulation == 2) { exactSol = GetExactSolutionValueTorus(xGauss); GetExactSolutionGradientTorus(xGauss, solGrad ); }

l2norm += (exactSol − soluGauss ) ∗ ( e x a c t S o l − soluGauss ) ∗ weight ;

f o r(unsignedj=0; j < dim ; j ++) { seminorm += ((soluGauss x [ j ] − solGrad[ j ]) ∗ ( soluGauss x [ j ] − solGrad[j ])) ∗ weight ; } } } NumericVector∗ norm vec ; norm vec = NumericVector:: build(). release (); norm vec−>i n i t (msh−>n processors(), 1 , false , AUTOMATIC) ;

norm vec−>set(iproc , l2norm); norm vec−>c l o s e ( ) ; l2norm = norm vec−>l1 norm ( ) ;

norm vec−>set(iproc , seminorm); norm vec−>c l o s e ( ) ; seminorm = norm vec−>l1 norm ( ) ;

d e l e t e norm vec ;

std : : pair < double, double > norm ; norm. first = sqrt(l2norm);

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norm.second = sqrt(seminorm); return norm ; }