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Regularity of Weakly Harmonic Maps of Surfaces

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Regularity of Weakly Harmonic Maps of Surfaces REGULARITY OF WEAKLY HARMONIC MAPS OF SURFACES PEYAM RYAN TABRIZIAN NOVEMBER 22, 2009 Abstract This paper is an introduction to the regularity of weakly harmonic maps of Riemannian manifolds, for people with a strong background in Riemannian geometry, but who never had an acquaintance with harmonic maps. In the first part, I exhibit the main definitions and properties one needs to know about harmonic and weakly harmonic maps of (Riemannian) manifolds. In the second and main part of the paper, I prove an important regularity theorem of weakly harmonic maps of surfaces, namely that, under mild extra hypotheses, if u is a weakly harmonic map of Riemannian surfaces then u is C1,α. The exposition will follow very closely the one of Frederic Helein’s Harmonic Maps, Conservation Laws, and Moving Frames [5]. Introduction The study of harmonic maps is a very beautiful subject, which blends Riemannian Geometry, PDEs, and Complex Analysis in an elegant way, with enormous applications in those subjects. Moreover, a lot of PDEs in nature (such as harmonic functions in Complex Analysis, or the geodesic equation on a manifold) can be written in terms of harmonic maps. This paper, which can also be called ’Harmonic maps in a nutshell’, has two main objectives. In the first section, I will introduce the basic definitions and main properties of harmonic maps and weakly harmonic maps, with a view toward what is to come in the second section. In this latter section, I will prove in detail a regularity theorem of weakly harmonic maps, which basically says that a weakly harmonic map, under very mild additional hypotheses, is more regular than one might expect. I will, in particular, introduce the use of Coulomb frames, which will prove to be an essential tool in the study of harmonic maps. Finally, let me acknowledge that this paper will very closely follow the book Harmonic Maps, Conservation Laws, and Moving Frames [5]. It is intended to be a paper in which one can learn about the main theorem in the book (presented in the second section) without having to read the whole book. 1 Harmonic maps: definition and main properties 1.1 Preliminaries For a good review of Riemannian geometry, consult Lee’s book Riemannian Geometry: An Introduction to Curvature [7]. M and N are smooth manifolds, supposed to be C1 (this is not a very restrictive definition, because one of Whitney’s theorems states that every C1-manifold is C1-diffeomorphic to a C1 manifold). M will be equipped with a C0,α Riemannian metric g, where 0 < α < 1. For N , there are two cases: either it is an abstract manifold with a C1-Riemannian metric h, or it is a C2-immersed submanifold of RN . The second case is especially useful, because by Nash’s theorem (see [8]), if h is Cl, for l ≥ 3, then there exists a Cl isometric immersion of (N ; h) in (RN ; h:; :i). Finally m := dim(M) and n := dim(N ). 1.2 Harmonic maps on surfaces There are essentially two ways to define harmonic functions on a Riemannian manifold. One is with help of coordinates: On (M; g), we can define an associated Laplacian operator ∆g acting on all smooth functions on M taking their values in R. Using a local coordinate system (x1; :::; xm) on M, denote by @ @ g (x) = g(x) ; αβ @xα @xβ the coefficients of the metric, which determines a matrix G(x), and let det(g(x)) := det(G(x)). Then, for each real-valued function φ defined over an open subset Ω of M, define 1 @ p @φ ∆ φ = p detg gαβ(x) (1) g detg @xα @xβ where we adopt Einstein’s convention, where repeated indices should be summed over. αβ Note. g (x) are the entries of the inverse of gαβ(x): Now we are ready to define harmonic functions: Definition. A smooth function φ defined over an open subset Ω of M and satisfying ∆gφ = 0 is called a harmonic function. Now although the above definition is very helpful when doing computations with harmonic functions, it does not lead any insight into their ’essence’. The following variational definition of a harmonic function does precisely this job. Define 1 p 1 m dvolg = detg(x)dx :::dx (2) be the Riemannian measure. For each smooth φ :Ω ⊆ M −! R, let Z E(Ω;g)(φ) = e(φ)dvolg (3) Ω be the energy or Dirichlet integral of φ (E may be finite or not), where e(φ) is the energy density of φ, given by 1 @φ @φ e(φ) = gαβ(x) (4) 2 @xα @xβ It is easy to check that the Dirichlet integral does not depend on the choice of the local coordinate system, and that, if is a compactly supported smooth function on Ω ⊂ M, then, 8t 2 R Z 2 E(Ω;g)(φ + t ) = E(Ω;g)(φ) − t (∆gφ) dvolg + O(t ) (5) Ω In this way, −∆g reappears as the variational derivative of EΩ, which provides us with an equivalent definition of the Laplacian. With this, we can make the following alternative definition of a harmonic function: Definition. A smooth function φ defined over an open subset Ω of M which is a critical point of its Dirichlet integral E(Ω;g)(φ) is called a harmonic function. Hence, using this variational definition, it is easy to see that the Laplacian does not depend on the co- ordinate system used. However, it depends on the metric used (more precisely, it depends on the conformal structure of M, see below). 1.3 Harmonic maps between two Riemannian manifolds Because we will be dealing with harmonic maps of surfaces, we need to generalize the definition of harmonic maps given previously to the case where the image of u is a Riemannian manifold as well. So let (N ; h) be a Riemannian manifold, compact and without boundary, and let u : M −! N be a smooth map. Definition. u is a harmonic map from (M; g) to (N ; h) if and only if u satisfies at each point x in M the equation: @uj @uk ∆ ui + gαβ(x)Γi (u(x)) = 0 (6) g jk @xα @xβ where 1 @h @h @h Γi = hil jl + kl − jk (7) jk 2 @xk @xk @xl are the Christoffel symbols. Similarly to before, harmonic maps are critical point of the Dirichlet functional Z E(M;g)(u) = e(u)(x)dvolg (8) M where 1 @ui @uj e(u)(x) = gαβ(x)h (u(x)) 2 ij @xα @xβ Here, we must interpret the notion of ’critical point’ in a specific sense: 2 Definition. A map u : M −! is a critical point of E(M;g) if and only if, for each one-parameter family of deformations ut : M −! N ; t 2 I ⊂ R 1 which has a C dependence on t, and is such that u0 ≡ u on M and, for every t, ut = u outside a compact subset K of M, we have E (ut) − E (u) lim (M;g) (M;g) = 0 t!0 t Again, all of the above definitions do not depend on the coordinates used, and the Dirichlet integral depends only on the conformal structure. 1.4 Conservation laws for harmonic maps Conservation laws are very important for the study of harmonic maps, in particular if there is a symmetry on the image manifold N , especially for the sphere S2. And, as we will see in the next main section, even in the absence of symmetries, those laws can be useful. For this, we need a couple of preliminary definitions. Let Ω be an open subset of M and let L be a Lagrangian defined for maps from Ω to N , that is, we suppose that L is a C1-function with values in R defined on: ∗ ∗ T N ⊗M×N T M := f(x; y; A) j (x; y) 2 M × N ;A 2 TyN ⊗ Tx Mg Note that in general, if V and W are vector spaces, then V ⊗ W ∗ =∼ Hom(W; V ), so in particular A i can be viewed as a linear map from TxM to TyN , and hence also as a matrix (in particular, Aα denotes the (i; α)th component of A). Now let dµ(x) be a C1 density measure on Ω. Then we can define a functional L on the set C1(Ω; N ) by letting Z L(u) = L(x; u(x); du(x))dµ(x) Ω If we are given metrics g and h over M and N , then we may choose 1 @ui @uj L(x; u(x); du(x)) = e(u)(x) = gαβ(x)h (u(x)) 2 ij @xα @xβ and dµ(x) = dvolg(x) which simplifies things a bit. Definition. Let X be a tangent vector field on N . Then X is an infinitesimal symmetry for L if and only if i @L i @L @X j i (x; y; A)X (y) + i (x; y; A) j Aα = 0 (9) @y @Aα @y This definition may be thought of as follows. Suppose that the vector field X is Lipschitz. Then, because N is compact, we may integrate the flow of X for all time. That is, for any y 2 N ; t 2 R, write expy(tX) = γ(t) 2 N where γ is the solution of γ(0) = y dγ (10) dt = X(γ) Hence, if X is an infinitesimal symmetry for L, then for every map u :Ω −! N , we have: L(x; expu(tX); d(expu(tX))) = L(x; u; du) (11) To check this, it suffices to differentiate equation (11) with respect to t and use (9).
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