University of Texas Rio Grande Valley ScholarWorks @ UTRGV Mathematical and Statistical Sciences Faculty Publications and Presentations College of Sciences 9-2019 Shape Equations for Two-Dimensional Manifolds through a Moving Frame Variational Approach Paul Bracken The University of Texas Rio Grande Valley, [email protected] Follow this and additional works at: https://scholarworks.utrgv.edu/mss_fac Part of the Mathematics Commons Recommended Citation Bracken, Paul. 2019. “Shape Equations for Two-Dimensional Manifolds through a Moving Frame Variational Approach.” International Journal of Geometric Methods in Modern Physics 16 (10): 1950155. https://doi.org/10.1142/S021988781950155X. This Article is brought to you for free and open access by the College of Sciences at ScholarWorks @ UTRGV. It has been accepted for inclusion in Mathematical and Statistical Sciences Faculty Publications and Presentations by an authorized administrator of ScholarWorks @ UTRGV. For more information, please contact [email protected], [email protected]. Shape Equations for Two-Dimensional Manifolds through a Moving Frame Variational Approach Paul Bracken Department of Mathematics, University of Texas, Edinburg, TX 78540, USA [email protected] Abstract A variational approach is considered which can be applied to functionals of a general form to determine a corresponding Euler-Lagrange or shape equation. It is the intention to formulate the theory in detail based on a moving frame approach. It is then applied to a functional of a general form which depends on both the mean and Gaussian curvatures as well as the area and volume elements of the manifold. Only the case of a two-dimensional closed manifold is considered. The first variation of the functional is calculated in terms of the variations of the basic variables of the manifold. The results of the first variation allow for the second variation of the functional to be evaluated. Keywords: metric, geometry, manifold, curvature, shape equations, variational, moving frame Mathematics Subject Classification: 49Q10, 53A05, 53Z05 1 Introduction A membrane with or without a freely exposed edge can be thought of as a surface or differentiable manifold of dimension two in three-dimensional Euclidean space. The case in which the surface is closed so the boundary is the empty set can be formulated as a variational problem [1]. The result of carrying this out is a mathematical description of a surface in the form of differential equations. Equations derived in this manner are usually referred to as shape equations. Exterior differential forms are introduced to describe the surface in terms of a set of structure equations [2 − 4]. This mathematical approach has been used to discuss the relationship between non-linear partial differential equations and the theory of surfaces in terms of their applications [5]. The functiomal which is introduced and studied has a very general structure of the form Z Z I = E2H(r);K(r) dA + p dV: (1.1) M V In (1.1), the functions H and K are the mean curvature and Gaussian curvature at point r of M and p is a constant, M refers to integration on the surface and V is the volume enclosed within the surface. The mean curvature is the average of the principal curvatures, and the Gaussian curvature is the product of these two curvatures. The function E in the integrand of (1.1) should be differentiable with respect to both slots. A specific form for (1.1) which has been studied already is the case in which E is proportional to the square of the mean curvature. This functional gives the Willmore energy or the classical binding energy which comes up in many applications in many areas of science as diverse as biology [6] and string theory [7]. Associated with this functional is the Willmore conjecture. This is one of the more prominent results in the study of Willmore energies and has been proved by Marques and Neves [8] by applying min-max theory of minimal surfaces. The energy expresses how much the surface has deviated from a round sphere. The minimizers of the Willmore functional are referred to as Willmore surfaces [9; 10]. These correspond to the solutions of the resulting Euler-Lagrange equation corresponding to (1.1) in this Willmore case [11]. This Willmore energy is important in the context of conformal geometry, since it is known to be invariant under the conformal transformations of E3. To put some of this in a more physical context, liquid crystal structures in physics can be 2 regarded as a smooth surface as well, with or without boundary depending on the nature of the system under consideration [12]. Examples of these appear in very distinct areas such as con- densed matter physics and biology [13]. Both lipid bilayers and cell membranes are referred to collectively as bio-membranes. The geometric formalism that is proposed here is very well suited to investigate the shapes and stability of such membranes. These objects might be described as closed surfaces, or bilayers with exposed edges, which in a variational framework can be treated as a boundary term. Physical quantities such as the total free energy can be assigned to these struc- tures. Theoretical study of such structures has the objectives of producing differential equations called shape equations by varying a given functional such as (1.1). The shape equations can be given in terms of a few parameters which can be used to model these kinds of physical systems. The intention here is to first introduce the necessary concept of the moving frame which appears in from differential geometry and necessary information pertaining to variational theory of two-dimensional manifolds. The variations of the all the differential forms have to be defined. The work is primarily devoted to the case of closed manifolds. The general functional (1.1) shall undergo a variational treatment to formulate an Euler-Lagrange equation in the basic surface variables. This allows the discussion of the equilibrium shapes and mechanical stabilities of closed surfaces and admit the type of application already mentioned. The first and second variations of functional (1.1) is obtained first. The first variation produces the Euler-Lagrange equation corresponding to the associated functional. These results can be used later to produce shape equations of various kinds which are relevant to the study of many physical systems [14 − 16]. 2 Surfaces in Three-Dimensional Euclidean Space Let M be a smooth orientable manifold in three-dimensional Euclidean space E3. At every point P of M an orthonormal system e1; e2; e3 can be constructed such that e3 is normal to the surface so they satisfy ei · ej = δij. The set fP; e1; e2; e3g is called a moving frame for the manifold. The difference between two frames at points P and P 0 very close to P is given as dr = !1e1 + !2e2; (2.1) 3 dei = !ijej; i = 1; 2; 3; (2.2) where !1;!2 and !ij (i; j = 1; 2; 3) are 1-forms. From ei · ej = δij it follows that !ij = −!ji, and from ddr = 0, the structure equations of the surface satisfied by these differential forms follow, d!1 = !12 ^ !2; (2.3) d!2 = !21 ^ !1; (2.4) !1 ^ !13 + !2 ^ !23 = 0; (2.5) d !ij = !ik ^ !kj; i; j = 1; 2; 3: (2.6) By Cartan's lemma, (2.5) implies that there exist functions on M such that !13 and !23 can be written !13 = a!1 + b!2;!23 = b!1 + c!2: (2.7) The area form on M which appears in (1.1) is given by d A = !1 ^ !2; (2.8) Using these basis differential forms, the three fundamental forms for the surface can be expressed as I = !1 ⊗!1 +!2 ⊗!2;II = a!1 ⊗!1 +2b!1 ⊗!2 +c!2 ⊗!2;III = !31 ⊗!31 +!32 ⊗!32: (2.9) In terms of the functions a, b and c, the mean curvature H and Gaussian curvature K of M are defined to be 1 H = (a + c);K = a c − b2: (2.10) 2 The operator ∗ will appear frequently. It is called the Hodge operator and it is defined on M so that if f and g are ordinary functions, then ∗f = f !1 ^ !2; ∗!1 = !2; ∗!2 = −!1: (2.11) 4 The operator ∗ can be used to define differential operators on M such as 2 d ∗ df = r f !1 ^ !2; (2.12) where r2 is the Laplace-Beltrami operator. With respect to a basis of one-forms !1;!2, the differential df is expressed as df = f1!1 + f2!2; (2.13) 2 which serves to define the derivatives fi. The Gauss map G : M ! S is defined as G(r) = e3(r), where S2 is the unit sphere. It induces a linear map denoted G∗ :Λ1 ! Λ1 of forms so that using (2.13), ∗ ∗ ∗ ∗ ∗ G !1 = !13; G !2 = !23 G df = f1G !1 + f2G !2: (2.14) Consequently, a new differential operator denoted d~ = G∗ d can be defined with respect to (2.13) as ~ df = f1!13 + f2!23: (2.15) To represent this more effectively, an operator ∗~ is defined such that ∗~!13 = !23 and ∗~!23 = −!13. Since d∗~df~ is a two form, there is a corresponding operator r · r~ defined such that ~ ~ d∗~df = r · rf !1 ^ !2: (2.16) The variational theory of surfaces which is required will be introduced. If we let M undergo an infinitesimal deformation such that every point r of M has a displacement d r, a new surface M 0 = fr0jr0 = r + δrgis obtained. The quantity δr is called the variation of the surface M and may be expressed as δr = δ1r + δ2r + δ3r; δir = Ωijei; (2.17) and the repeated index is not summed in (2.17).