DOCSLIB.ORG

HARMONIC MAPPINGS BETWEEN RIEMANNIAN MANIFOLDS by Anand Arvind Joshi a Thesis Presented to the FACULTY of the GRADUATE SCHOOL UN

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
HARMONIC MAPPINGS BETWEEN RIEMANNIAN MANIFOLDS by Anand Arvind Joshi a Thesis Presented to the FACULTY of the GRADUATE SCHOOL UN HARMONIC MAPPINGS BETWEEN RIEMANNIAN MANIFOLDS by Anand Arvind Joshi A Thesis Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree MASTER OF ARTS (MATHEMATICS) August 2006 Copyright 2006 Anand Arvind Joshi Dedication I dedicate this thesis to my parents Arvind and Suvarna Joshi. ii Table of Contents Dedication ii Abstract iv 1 Harmonic Mappings 1 1.1 SpaceofMaps............................... 1 1.2 ConnectionsintheSpaceofMaps . 4 1.3 TheFirstVariationFormula. 11 1.4 HarmonicMaps.............................. 16 2 The Heat Flow Method 18 2.1 TheEellsSampsonTheorem . 18 2.2 TheHeatFlowMethod .......................... 19 3 Existence of Local and Global Solutions 31 3.1 ExistenceofLocalSolutions . 31 3.2 ExistenceofGlobalTime-DependentSolutions . ..... 43 Bibliography 48 Appendices 49 A Holder¨ Spaces and Some Results on PDEs 49 A.1 H¨olderSpaces............................... 49 A.2 SomeResultsonPDEs .......................... 51 A.3 Fr´echetDerivative.. .. .. .. .. .. .. .. 53 iii Abstract Harmonic mappings between two Riemannian manifolds is an object of extensive study, due to their wide applications in mathematics, science and engineering. Proving the existence of such mappings is challenging because of the non-linear nature of the cor- responding partial differential equations. This thesis is an exposition of a theorem by Eells and Sampson, which states that any given map from a Riemannian manifold to a Riemannian manifold with non-positive sectional curvature can be freely homotoped to a harmonic map. In particular, this proves the existence of harmonic maps between such manifolds. The technique used for the proof is the heat-flow method. iv Chapter 1 Harmonic Mappings In this chapter we define and discuss harmonic mappings. Let (M,g) and (N, h) be m and n dimensional Riemannian manifolds, and let u denote a smooth map from M to N, i.e. u ∈ C∞(M, N). A natural question to ask is: what is the ‘least expanding’ map from M to N? In order to make precise what we mean by ‘least expanding’ map here, we need to analyze the space of maps C∞(M, N). In the sections that follow, we do this analysis and define an energy of maps on this space. A harmonic map will be a critical point of this energy as discussed later. 1.1 SpaceofMaps ∗ Let TxM denote the tangent space of M and let TxM be the dual space of this tan- ∞ gent space. We know that u ∈ C (M, N) implies that dux is a linear map from TxM to Tu(x)N, i.e. dux ∈ Hom(TxM, Tu(x)N). We want to find a metric on Hom(TxM, Tu(x)N) so that we can define energy of maps. We first prove a lemma. ∼ ∗ Lemma 1.1.1. TxM = TxM . The isomorphism is linear and induces a metric on ∗ TxM . 1 Proof. The Riemannian metric g induces a natural linear isomorphism between TxM ∗ m i ∂ and its dual TxM defined as follows. Let Xx = i=0 X (x)( ∂xi )x ∈ TxM and wx = m i ∗ ∗ i=0 wi(x)(dx )x ∈ TxM . Define ♭ : TxM → TxPM and ♯ : TxM → TxM P m m ♭ j i Xx = ( gij(x)X (x))(dx )x, (1.1) i=1 i=1 X X m m ∂ w♯ = ( gij(x)w (x))( ) . (1.2) j ∂xi x i=1 i=1 X X Clearly ♯ and ♭ are linear. Also it can be verified that they are inverse of each other ∗ ∗ resulting in a linear isomorphism between TxM and TxM . Now we define a metric gx ∗ on TxM by ∗ ♯ ♯ ∗ gx(wx, θx)= gx(wx, θx) for wx, θx ∈ TxM . (1.3) ∗ ∗ i j This bilinear form gx is a metric due to linearity of ♯. We can also get gx((dx)x, (dx)x)= ij ij gx where (g ) denotes the matrix inverse of g =(gij). ∼ ∗ Proposition 1.1.2. Hom(TxM, Tu(x)N) = TxM ⊗ Tu(x)N. † ∗ Proof. For every f ∈ Hom(TxM, Tu(x)N) we associate a bilinear map f ∈ TxM ⊗ Tu(x)N by defining † ∗ f (V,w)= w(f(V )), ∀V ∈ TxM,w ∈ Tu(x)N . (1.4) We can see that, given such a bilinear map, we can also associate with it a linear map in Hom(TxM, Tu(x)). We know that dux ∈ Hom(TxM, Tu(x)N) is represented in local coordinates by ∂ m ∂uα ∂ du = (x) . (1.5) x ∂xi ∂xi ∂yα α=1 u(x) X 2 ∗ Since the basis for TxM ⊗ Tu(x)N is given by ∂ (dxi) ⊗ ( ) , (1.6) x ∂yα u(x) dux is represented by m n ∂uα ∂ du = (x)(dxi) ⊗ . (1.7) x ∂xi x ∂yα i=1 α=1 u(x) X X ij ∗ We proved that g is an inner product on TxM . Also hu(x) is the induced inner ∗ product in Tu(x). These two inner products induce an inner product on TxM ⊗ Tu(x)N given by ∂ ∂ (dxi) ⊗ , (dxj) ⊗ = gijh (u(x)). (1.8) x ∂yα x ∂yβ αβ * u(x) u(x)+ Since this inner product is defined everywhere on M, we define an inner product on sections by setting ′ ′ ′ ∗ −1 hσ, σ i(x)= hσ(x), σ (x)ix, for x ∈ M; σ, σ ∈ Γ(T M ⊗ u T N). (1.9) With this inner product, we define a norm on dux given by m n ∂uα ∂uβ |du|2 = gijh (u) . (1.10) αβ ∂xi ∂xj i,j=1 X α,βX=1 With this norm defined on Hom(TM,TN), we now define the energy density of a map. Definition 1.1.1. Given u ∈ C∞(M, N), the energy density function of u is defined as 1 e(u)(x)= |du|2 (x), x ∈ M. (1.11) 2 3 Definition 1.1.2. Let (M,g) be a compact Riemannian manifold. Given u ∈ C∞(M, N), the energy or harmonic energy of u is defined as 1 E(u)= e(u)dµ = |dµ |. (1.12) g 2 g ZM ZM The energy density of u can be interpreted in a following way. Let {e1, ..., em}, and ′ ′ {e1, ..., en} be orthonormal bases with respect to gx and hu(x), for tangent spaces TxM and Tu(x)N, respectively. We express dux in these bases as n α ′ dux(ei)= λi eα, i =1, ..., m. (1.13) α=1 X Then, we get m n 2 α 2 |du| (x)= (λi ) . (1.14) i=1 α=1 X X Consequently, we can regard the energy density functional e(u)(x) as the ‘rate of expan- sion’ of the differential dux : TxM → Tu(x)N of u at x ∈ M. This is why we call e(ux) ‘energy density’ of the map. Thus the energy E(u) is defined for each u ∈ C∞(M, N). The energy of maps E can be regarded as a functional E : C∞(M, N) → R, and we want to find maps which are critical points of this functional E. 1.2 Connections in the Space of Maps Having introduced an inner product and norm for u ∈ Hom(M, N) i.e. on Γ(T M ∗ ⊗ u−1T N) we want to know what is the effect on energy if the map u is changed by a small amount. In other words, we want to be able to take directional derivatives in the 4 space Γ(T M ∗ ⊗u−1T N). To do that, we develop the notion of connection for this space. First, we develop connections for T M ∗ and u−1T N. Let ∇ denote the Levi-Civita connection of M which gives us a map ∇ : Γ(T M) → Γ(T M ∗ ⊗ T N) (1.15) which assigns a tensor field ∇Y ∈ Γ(T M ∗ ⊗ T M) of type (1, 1) to Y ∈ Γ(T M), a (0, 1) tensor field. ∇Y is the covariant differential of Y . Let ♯ and ♭ be the isomorphisms between T M and T M ∗ given in (1.1) and (1.2). We can define a connection ∇∗ in T M ∗ by setting ∗ ♯ ♭ ∗ ∇X w(Y )=(∇X w ) (Y ), Y ∈ Γ(T M),w ∈ Γ(T M ) (1.16) ♯ = gx(∇X w ,Y ) (1.17) ♯ ♯ = X(gx(w ,Y )) − gx(w , ∇X Y ) by compatibility of ∇ (1.18) = Xw(Y ) − w(∇X Y ) (1.19) which could also serve as an alternate definition for the connection ∇∗ and also explains that the connection ∇∗ on T M ∗ and connection ∇ on T M can be regarded as dual to each other. ∗ Because of the compatibility of ∇ with gij, we can see that the connection ∇ is compatible with the metric gij on T M ∗ by the following computation. ∗ ∗ ∗ ∗ ∗ Lemma 1.2.1. Xg (w, θ)= g (∇X w, θ)+ g (w, ∇X , θ). 5 Proof. ∗ ♯ ♯ ♯ ∗ ♯ RHS = g((∇X w) , θ )+ g(w , (∇X θ) ) (1.20) ♯ ♯ ♯ ♯ = g(∇X w , θ )+ g(w , ∇X θ ) (1.21) = Xw(θ♯)= LHS (1.22) Thus ∇∗ is a Riemannian connection. Now that we have introduced the connection ∇∗ in T M ∗, what are the connection coefficients? We use (1.19) to do the computation. ∗ k ∂ ∂ k ∂ k ∂ ∂ ∂ (1.23) (∇ dx )( l )= j dx ( l ) − dx (∇( ) l ) ∂xj ∂x ∂x ∂x ∂xj ∂x ∂ = δk − Γk (1.24) ∂xj l jl k = −Γjl (1.25) Thus we get an expression for ∇∗ as m ∗ k k j ∇ ∂ dx = − Γijdx , 1 ≤ i, k ≤ m. (1.26) ∂xi j=1 X We note that the connection coefficients of ∇∗ induced in T M ∗ from ∇ are negative of the connection coefficients of ∇. Now consider the tangent bundle u−1T N ⊂ T M induced from T N by the map u : M → N. At each point x, ∂ ∂ ◦ u (x), ..., ◦ u (x) (1.27) ∂y1 ∂yn 6 −1 gives rise to a base for the fiber Tu(x)N of u T N over x.
Load more

Recommended publications
  • 1.10 Partitions of Unity and Whitney Embedding 1300Y Geometry and Topology
    1.10 Partitions of unity and Whitney embedding 1300Y Geometry and Topology Theorem 1.49 (noncompact Whitney embedding in R2n+1). Any smooth n-manifold may be embedded in R2n+1 (or immersed in R2n). Proof. We saw that any manifold may be written as a countable union of increasing compact sets M = [Ki, and that a regular covering f(Ui;k ⊃ Vi;k;'i;k)g of M can be chosen so that for fixed i, fVi;kgk is a finite ◦ ◦ cover of Ki+1nKi and each Ui;k is contained in Ki+2nKi−1. This means that we can express M as the union of 3 open sets W0;W1;W2, where [ Wj = ([kUi;k): i≡j(mod3) 2n+1 Each of the sets Ri = [kUi;k may be injectively immersed in R by the argument for compact manifolds, 2n+1 since they have a finite regular cover. Call these injective immersions Φi : Ri −! R . The image Φi(Ri) is bounded since all the charts are, by some radius ri. The open sets Ri; i ≡ j(mod3) for fixed j are disjoint, and by translating each Φi; i ≡ j(mod3) by an appropriate constant, we can ensure that their images in R2n+1 are disjoint as well. 0 −! 0 2n+1 Let Φi = Φi + (2(ri−1 + ri−2 + ··· ) + ri) e 1. Then Ψj = [i≡j(mod3)Φi : Wj −! R is an embedding. 2n+1 Now that we have injective immersions Ψ0; Ψ1; Ψ2 of W0;W1;W2 in R , we may use the original argument for compact manifolds: Take the partition of unity subordinate to Ui;k and resum it, obtaining a P P 3-element partition of unity ff1; f2; f3g, with fj = i≡j(mod3) k fi;k.
    [Show full text]
  • Math 865, Topics in Riemannian Geometry
    Math 865, Topics in Riemannian Geometry Jeff A. Viaclovsky Fall 2007 Contents 1 Introduction 3 2 Lecture 1: September 4, 2007 4 2.1 Metrics, vectors, and one-forms . 4 2.2 The musical isomorphisms . 4 2.3 Inner product on tensor bundles . 5 2.4 Connections on vector bundles . 6 2.5 Covariant derivatives of tensor fields . 7 2.6 Gradient and Hessian . 9 3 Lecture 2: September 6, 2007 9 3.1 Curvature in vector bundles . 9 3.2 Curvature in the tangent bundle . 10 3.3 Sectional curvature, Ricci tensor, and scalar curvature . 13 4 Lecture 3: September 11, 2007 14 4.1 Differential Bianchi Identity . 14 4.2 Algebraic study of the curvature tensor . 15 5 Lecture 4: September 13, 2007 19 5.1 Orthogonal decomposition of the curvature tensor . 19 5.2 The curvature operator . 20 5.3 Curvature in dimension three . 21 6 Lecture 5: September 18, 2007 22 6.1 Covariant derivatives redux . 22 6.2 Commuting covariant derivatives . 24 6.3 Rough Laplacian and gradient . 25 7 Lecture 6: September 20, 2007 26 7.1 Commuting Laplacian and Hessian . 26 7.2 An application to PDE . 28 1 8 Lecture 7: Tuesday, September 25. 29 8.1 Integration and adjoints . 29 9 Lecture 8: September 23, 2007 34 9.1 Bochner and Weitzenb¨ock formulas . 34 10 Lecture 9: October 2, 2007 38 10.1 Manifolds with positive curvature operator . 38 11 Lecture 10: October 4, 2007 41 11.1 Killing vector fields . 41 11.2 Isometries . 44 12 Lecture 11: October 9, 2007 45 12.1 Linearization of Ricci tensor .
    [Show full text]
  • Harmonic Maps from Surfaces of Arbitrary Genus Into Spheres
    Harmonic maps from surfaces of arbitrary genus into spheres Renan Assimos and J¨urgenJost∗ Max Planck Institute for Mathematics in the Sciences Leipzig, Germany We relate the existence problem of harmonic maps into S2 to the convex geometry of S2. On one hand, this allows us to construct new examples of harmonic maps of degree 0 from compact surfaces of arbitrary genus into S2. On the other hand, we produce new example of regions that do not contain closed geodesics (that is, harmonic maps from S1) but do contain images of harmonic maps from other domains. These regions can therefore not support a strictly convex function. Our construction builds upon an example of W. Kendall, and uses M. Struwe's heat flow approach for the existence of harmonic maps from surfaces. Keywords: Harmonic maps, harmonic map heat flow, maximum principle, convexity. arXiv:1910.13966v2 [math.DG] 4 Nov 2019 ∗Correspondence: [email protected], [email protected] 1 Contents 1 Introduction2 2 Preliminaries4 2.1 Harmonic maps . .4 2.2 The harmonic map flow . .6 3 The main example7 3.1 The construction of the Riemann surface . .7 3.2 The initial condition u0 ............................8 3.3 Controlling the image of u1 ......................... 10 4 Remarks and open questions 13 Bibliography 15 1 Introduction M. Emery [3] conjectured that a region in a Riemannian manifold that does not contain a closed geodesic supports a strictly convex function. W. Kendall [8] gave a counterexample to that conjecture. Having such a counterexample is important to understand the relation between convexity of domains and convexity of functions in Riemannian geometry.
    [Show full text]
  • Simple Stable Maps of 3-Manifolds Into Surfaces
    Tqmlogy. Vol. 35, No. 3, pp. 671-698. 1996 Copyright 0 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved 004@9383,96/315.00 + 0.00 0040-9383(95)ooo3 SIMPLE STABLE MAPS OF 3-MANIFOLDS INTO SURFACES OSAMU SAEKI~ (Receioedjiir publication 19 June 1995) 1. INTRODUCTION LETS: M3 -+ [w2 be a C” stable map of a closed orientable 3-manifold M into the plane. It is known that the local singularities off consist of three types: definite fold points, indefinite fold points and cusp points (for example, see [l, 91). Note that the singular set S(f) offis a smooth l-dimensional submanifold of M. In their paper [l], Burlet and de Rham have studied those stable maps which have only definite fold points as the singularities and have shown that the source manifolds of such maps, called special generic maps, must be diffeomorphic to the connected sum of S3 and some copies of S’ x S2 (see also [14,16]). On the other hand, Levine [9] has shown that we can eliminate the cusp points of any stable mapfby modifying it homotopically; i.e., every orientable 3-manifold admits a stable map into the plane without cusp points. In this paper we investigate an intermediate class of stable maps, namely simple ones [ 171. A stable mapf: M -+ R2 is simple iffhas no cusp points and if every component of the fiberf -l(x) contains at most one singular point for all x E [w’.In the terminology of [S, lo], fis simple if and only if it has no vertices.
    [Show full text]
  • A New Explicit Way of Obtaining Special Generic Maps Into the 3
    A NEW EXPLICIT WAY OF OBTAINING SPECIAL GENERIC MAPS INTO THE 3-DIMENSIONAL EUCLIDEAN SPACE NAOKI KITAZAWA Abstract. A special generic map is a smooth map regarded as a natural gen- eralization of Morse functions with just 2 singular points on homotopy spheres. Canonical projections of unit spheres are simplest examples of such maps and manifolds admitting special generic maps into the plane are completely deter- mined by Saeki in 1993 and ones admitting such maps into general Euclidean spaces are determined under appropriate conditions. Moreover, if the difference of dimensions of source and target manifolds are not so large, then the diffeomorphism types of source manifolds are often limited; for example, homotopy spheres except standard spheres do not admit special generic maps into Euclidean spaces whose dimensions arenot so low. As another example, 4-dimensional simply connected manifolds admitting special generic maps are only manifolds represented as the connected sum of the total spaces of 2-dimensional sphere bundles over the 2-dimensional sphere (or the 4-dimensional standard sphere) and there are many manifolds homeomorphic and not diffeomorphic to these manifolds. These explicit facts make special generic maps attractive objects in the theory of Morse functions and higher dimensional versions and application to algebraic and differentiable topology of manifolds, which is an important study in both singuarity theory of maps and algebraic and differential topology of manifolds. In this paper, we demonstrate a way of construction of special generic maps into the 3-dimensional Euclidean space. For this, first we prepare maps onto 2-dimensional polyhedra regarded as simplicial maps naturally called pseudo quotient maps, which are generalizations of the quotient maps to the spaces of all the connected components of inverse images, so-called Reeb spaces of original smooth maps, being fundamental and important tools in the studies.
    [Show full text]
  • Superrigidity, Generalized Harmonic Maps and Uniformly Convex Spaces
    SUPERRIGIDITY, GENERALIZED HARMONIC MAPS AND UNIFORMLY CONVEX SPACES T. GELANDER, A. KARLSSON, G.A. MARGULIS Abstract. We prove several superrigidity results for isometric actions on Busemann non-positively curved uniformly convex metric spaces. In particular we generalize some recent theorems of N. Monod on uniform and certain non-uniform irreducible lattices in products of locally compact groups, and we give a proof of an unpublished result on commensurability superrigidity due to G.A. Margulis. The proofs rely on certain notions of harmonic maps and the study of their existence, uniqueness, and continuity. Ever since the first superrigidity theorem for linear representations of irreducible lat- tices in higher rank semisimple Lie groups was proved by Margulis in the early 1970s, see [M3] or [M2], many extensions and generalizations were established by various authors, see for example the exposition and bibliography of [Jo] as well as [Pan]. A superrigidity statement can be read as follows: Let • G be a locally compact group, • Γ a subgroup of G, • H another locally compact group, and • f :Γ → H a homomorphism. Then, under some certain conditions on G, Γ,H and f, the homomorphism f extends uniquely to a continuous homomorphism F : G → H. In case H = Isom(X) is the group of isometries of some metric space X, the conditions on H and f can be formulated in terms of X and the action of Γ on X. In the original superrigidity theorem [M1] it was assumed that G is a semisimple Lie group of real rank at least two∗ and Γ ≤ G is an irreducible lattice.
    [Show full text]
  • Solving Variational Problems and Partial Differential Equations Mapping Into General Target Manifolds
    Journal of Computational Physics 195 (2004) 263–292 www.elsevier.com/locate/jcp Solving variational problems and partial differential equations mapping into general target manifolds Facundo Memoli a, Guillermo Sapiro a,*, Stanley Osher b a Electrical and Computer Engineering, University of Minnesota, 200 Union Street, Minneapolis, MN 55455, USA b UCLA Mathematics Department, Los Angeles, CA 90095, USA Received 6 May 2003; received in revised form 30 September 2003; accepted 4 October 2003 Abstract A framework for solving variational problems and partial differential equations that define maps onto a given ge- neric manifold is introduced in this paper. We discuss the framework for arbitrary target manifolds, while the domain manifold problem was addressed in [J. Comput. Phys. 174(2) (2001) 759]. The key idea is to implicitly represent the target manifold as the level-set of a higher dimensional function, and then implement the equations in the Cartesian coordinate system where this embedding function is defined. In the case of variational problems, we restrict the search of the minimizing map to the class of maps whose target is the level-set of interest. In the case of partial differential equations, we re-write all the equationÕs geometric characteristics with respect to the embedding function. We then obtain a set of equations that, while defined on the whole Euclidean space, are intrinsic to the implicitly defined target manifold and map into it. This permits the use of classical numerical techniques in Cartesian grids, regardless of the geometry of the target manifold. The extension to open surfaces and submanifolds is addressed in this paper as well.
    [Show full text]
  • Reference Ic/86/6
    REFERENCE IC/86/6 INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS a^. STABLE HARMONIC MAPS FROM COMPLETE .MANIFOLDS Y.L. Xin INTERNATIONAL ATOMIC ENERGY AGENCY UNITED NATIONS EDUCATIONAL, SCIENTIFIC AND CULTURAL ORGANIZATION 1986 Ml RAM ARE-TRIESTE if ic/66/6 I. INTRODUCTION In [9] the author shoved that there is no non-.^trioLHri-t Vuliip harmonic maps from S with n > 3. Later Leung and Peng ['i] ,[T] proved a similar result for stable harmonic maps into S . These results remain true when the sphere is replaced by certain submanifolds in the Euclidean space [5],[6]. It should be International Atomic Energy Agency pointed out that these results are valid for compact domain manifolds. Recently, and Schoen and Uhlenbeck proved a Liouville theorem for stable harmonic maps from United Nations Educational Scientific and Cultural Organization Euclidean space into sphere [S]. The technique used In this paper inspires us INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS to consider stable harmonic maps from complete manifolds and obtained a generalization of previous results. Let M be a complete Riemannian manifold of dimension m. Let N be an immersed submanifold of dimension n in ]Rn * with second fundamental form B. Consider a symmetric 2-tensor T = <B(-,e.), B( •,£.)> in N, where {c..} Is a local orthonormal frame field. Here and in the sequel we use the summation STABLE HAKMOIIIC MAPS FROM COMPLETE MANIFOLDS * convention and agree the following range of Indices: Y.L. Xin ** o, B, T * n. International Centre for Theoretical Physics, Trieste, Italy. We define square roots of eigenvalues X. of the tensor T to be generalized principal curvatures of S in TEn <1.
    [Show full text]
  • Harmonic Maps and Soliton Theory
    Harmonic maps and soliton theory Francis E. Burstall School of Mathematical Sciences, University of Bath Bath, BA2 7AY, United Kingdom From "Matematica Contemporanea", 2, (1992) 1-18 1 Introduction The study of harmonic maps of a Riemann sphere into a Lie group or, more generally, a symmetric space has been the focus for intense research by a number of Differential Geometers and Theoretical Physicists. As a result, these maps are now quite well understood and are seen to correspond to holomorphic maps into some (perhaps infinite-dimensional) complex manifold. For more information on this circle of ideas, the Reader is referred to the articles of Guest and Valli in these proceedings. In this article, I wish to report on recent progress that has been made in under- standing harmonic maps of 2-tori in symmetric spaces. This work has its origins in the results of Hitchin [9] on harmonic tori in S3 and those of Pinkall-Sterling [13] on constant mean curvature tori in R3 (equivalently, harmonic tori in S2). In those papers, the problem again reduces to one in Complex Analysis or, more properly, Algebraic Geometry but of a rather different kind: here the fundamental object is an algebraic curve, a spectral curve, and the equations of motion reduce to a linear flow on the Jacobian of this curve. This is a familiar situation in the theory of com- pletely integrable Hamiltonian systems (see, for example [1, 14]) and the link with Hamiltonian systems was made explicit in the study of Ferus-Pedit-Pinkall-Sterling [7] who showed that the non-superminimal minimal 2-tori in S4 arise from solving a pair of commuting Hamiltonian flows in a suitable loop algebra.
    [Show full text]
  • DIFFERENTIABLE MANIFOLDS Course C3.1B 2012 Nigel Hitchin
    DIFFERENTIABLE MANIFOLDS Course C3.1b 2012 Nigel Hitchin [email protected] 1 Contents 1 Introduction 4 2 Manifolds 6 2.1 Coordinate charts . .6 2.2 The definition of a manifold . .9 2.3 Further examples of manifolds . 11 2.4 Maps between manifolds . 13 3 Tangent vectors and cotangent vectors 14 3.1 Existence of smooth functions . 14 3.2 The derivative of a function . 16 3.3 Derivatives of smooth maps . 20 4 Vector fields 22 4.1 The tangent bundle . 22 4.2 Vector fields as derivations . 26 4.3 One-parameter groups of diffeomorphisms . 28 4.4 The Lie bracket revisited . 32 5 Tensor products 33 5.1 The exterior algebra . 34 6 Differential forms 38 6.1 The bundle of p-forms . 38 6.2 Partitions of unity . 39 6.3 Working with differential forms . 41 6.4 The exterior derivative . 43 6.5 The Lie derivative of a differential form . 47 6.6 de Rham cohomology . 50 2 7 Integration of forms 57 7.1 Orientation . 57 7.2 Stokes' theorem . 62 8 The degree of a smooth map 68 8.1 de Rham cohomology in the top dimension . 68 9 Riemannian metrics 76 9.1 The metric tensor . 76 9.2 The geodesic flow . 80 10 APPENDIX: Technical results 87 10.1 The inverse function theorem . 87 10.2 Existence of solutions of ordinary differential equations . 89 10.3 Smooth dependence . 90 10.4 Partitions of unity on general manifolds . 93 10.5 Sard's theorem (special case) . 94 3 1 Introduction This is an introductory course on differentiable manifolds.
    [Show full text]
  • SOME RESULTS of F-BIHARMONIC MAPS 1. Introduction Harmonic Maps Play a Central Roll in Variational Problem
    SOME RESULTS OF F -BIHARMONIC MAPS YINGBO HAN and SHUXIANG FENG Abstract. In this paper, we give the notion of F -biharmonic maps, which is a generalization of bi- harmonic maps. We derive the first variation formula which yields F -biharmonic maps. Then we investigate the harmonicity of F -biharmonic maps under the curvature conditions on the target mani- fold (N; h). We also introduce the stress F -bienergy tensor SF;2. Then, by using the stress F -bienergy tensor SF;2, we obtain some nonexistence results of proper F -biharmonic maps under the assumption that SF;2 = 0. Moreover, we derive some monotonicity formulas for the special case of the biharmonic map, i.e., where F -biharmonic map with F (t) = t. Then, by using these monotonicity formulas, we obtain new results on the non existence of proper biharmonic isometric immersions from complete manifolds. 1. Introduction Harmonic maps play a central roll in variational problems for smooth maps between manifolds 1 R 2 u:(M; g) ! (N; h) as the critical points of the energy functional E(u) = 2 M kduk dvg. On the other hand, in 1981, J. Eells and L. Lemaire [7] proposed the problem to consider the k-harmonic JJ J I II maps which are critical maps of the functional Z k 2 Go back k(d + δ) uk Ek(u) = dvg M 2 Full Screen Close Received January 10, 2013; revised July 15, 2013. 2010 Mathematics Subject Classification. Primary 58E20, 53C21. Key words and phrases. F -biharmonic map; stress F -bienergy tensor; harmonic map.
    [Show full text]
  • ON the VACUUM STRUCTURE in the COULOMB 1. Introduction
    Nuclear Physics B175 (1980) 526-5416 © North-Holland Publishing Company ON THE VACUUM STRUCTURE IN THE COULOMB AND LANDAU GAUGES Antti NIEMI 1 Helsinki University, Research Institute for Theoretical Physics, Siltavuorenpenger 20, 00170 Helsinki 17, Finland Received 5 May 1980 (Final version received 4 June 1980) Vacuum structure in the SU(N) Coulomb and Landau gauges is studied by using the methods of harmonic maps. A systematic way for solving the Gribov vacuum copy equation N presented and many examples are discussed in both the Coulomb and Landau gauges as applications of the method. Finally, the physical interpretation of Gribov ambiguities is shortly reviewed from a topological point of view. 1. Introduction In this article I present a systematic method of solving the Gribov vacuum copy equation in the SU(N) Coulomb and Landau gauges and discuss some properties of the allowed field configurations. The problem of Gribov ambiguities goes back to 1977 when Gribov [1, 2] observed that the Coulomb gauge-fixing condition does not uniquely fix the gauge but allows some remaining gauge freedom. After Gribov's observation an extensive literature has grown on the subject but so far an acceptable physical interpretation has not been given. Many configurations attainable in the vacuum sector are known [1-11]; the approach has been in terms of some more or less successful ansiitze and no profound relationships between different ansMze have been found. Mathematically the vacuum sector solutions of the gauge copy equation are harmonic maps between the manifolds R k and SU(N) [9], and in the present article I will use this observation to develop a systematic method for solving the guage copy equation in the vacuum sector.
    [Show full text]