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 C∞ (this is not a very restrictive deﬁnition, because one of Whitney’s theorems states that every C1-manifold is C1-diffeomorphic to a C∞ 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 = √ 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 ψ is a compactly supported smooth function on Ω ⊂ M, then, ∀t ∈ R Z 2 E(Ω,g)(φ + tψ) = E(Ω,g)(φ) − t (∆gφ)ψdvolg + 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 coordinate 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 ∈ 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 := {(x, y, A) | (x, y) ∈ M × N ,A ∈ TyN ⊗ Tx M} 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 ∈ N , t ∈ R, write

expy(tX) = γ(t) ∈ 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). The importance of inﬁnitesimal symmetries is due to the following theorem, called Noether’s theorem:

3 Theorem 1 (Noether’s theorem). Let X be a Lipschitz tangent vector ﬁeld on N , which is an inﬁnitesimal symmetry for L. If u :Ω −→ N is a critical point of L, where Z L(u) = L(x, u(x), du(x))dµ(x) Ω and dµ(x) is a C1-density measure on Ω, then:

∂L div .X(u) = 0 (12) ∂A Equivalently, using the coordinates (x1, ..., xm) on Ω such that dµ = ρ(x)dx1...dxm,

m X ∂ ∂L ρ(x)Xi(u) (x, u, du) = 0 (13) ∂xα ∂Ai α=1 α

This produces a divergence-free vector field over Ω. Consult [9] for more information. The importance of divergence-free vector fields in general stems from the fact that, by the divergence theorem, the integral over a divergence vector field depends only on its values on ∂Ω.

1.5 Weakly harmonic maps This concept generalizes the notion of harmonic maps between Riemannian manifolds and will be used throughout the remainder of the paper. As a matter of convenience, assume that (N , h) is isometrically embedded in (R, h., .i) (for the general case, see [3]). This allows us to use the following definition: Definition. Z 1 1 N 2 H (M, N ) = u ∈ Lloc(M, R ) | |du| dvolg < ∞, u(x) ∈ N a.e. (14) M The main problem is that, if M has dimension greater than or equal to 2, maps in H1(M, N ) are not smooth in general. Moreover, H1(M, N ) does not have a smooth manifold structure. This implies that different non-equivalent generalizations of the notion of harmonic maps exist in H1(M, N ). The one most 2 suitable for our purposes is the following. In order to state this definition, let N be of class C , Vδ(N ) be a N tubular neighborhood of radius δ of N in R , and P the projection from Vδ(N ) onto N defined such that P restricted to N is the identity map. Also, suppose δ is sufficiently small for P to be defined and C1. Now we can finally define our notion of a weakly harmonic map:

1 1 N Deﬁnition. We say that u ∈ H (M, N ) is weakly harmonic if and only if, for any map v in H0 (M, R ) ∩ ∞ N 1 N ∞ N 1 N L (M, R ) (where H0 (M, R ) is the closure of Cc (M, R ) in H (M, R )), we have: E (P (u + tv)) − E (u) lim (M,g) (M,g) = 0 (15) t→0 t

∞ N Note. (15) is well-defined for sufficiently small t, since v ∈ L (M, R ), and thus u + tv belongs to VδN −1 if |t| < δ kvkL∞ . For a more coordinate-dependent formula, one can check the following: Lemma 2. Suppose N is C2. Then, every weakly harmonic map u ∈ H1(M, N ) satisfies the equation

−∆g(u)⊥Tu(N ) 1 N ∞ N weakly, i.e. ∀v ∈ H0 (M, R ) ∩ L (M, R ), if v(x)⊥Tu(x)N a.e., then: Z αβ ∂u ∂v g (x) α , β dvolg = 0 (16) M ∂x ∂x

4 The following lemma characterizes the extent to which our deﬁnition depends on the isometric immersion used for (N , h). It will be useful for later on.

N1 N2 Lemma 3. Let J1 :(N , h) → N1 ⊂ R , and let J2 :(N , h) → N2 ⊂ R be two isometric immersions N1 N2 −1 of (N , h). Let Φ: R → R be the extension of J2 ◦ J1 : N1 → N2 deﬁned by: −1 Φ(y) = η(dist(y, N1))J2 ◦ J1 ◦ P (y) y ∈ V2δN1 dγ (17) dt = X(γ) y∈ / V2δN1 2 1 1 Then Φ is C , and u ∈ H (M, N1) is weakly harmonic if and only if Φ ◦ u ∈ H (M, N2) is weakly harmonic.

1.6 Examples of harmonic maps This section essentially relieves the dullness of this paper by showing that harmonic maps do exist in nature, and that a lot of PDEs in geometry can be reduced to the study of harmonic maps.

1.6.1 Example 1: RN -valued maps A map u :(M, g) → RN is harmonic if and only if each of its components ui is a real-valued harmonic function on (M, g). So, essentially, all of complex analysis, which is the study of real-valued harmonic functions, is embedded in the study of harmonic maps!

1.6.2 Example 2: Geodesics

If M has dimension 1 (i.e. it is either an interval in R or a circle) with variable t, then (16) becomes: d2u du du + A(u) , = 0 dt2 dt dt (the notation is as in Lemma 2), which is the equation satisﬁed by a constant speed parametrization of a geodesic in (N , h)!

1.6.3 Example 3: Maps taking their values in S2 In this case, we have N = S2, the unit sphere in R3 with its usual norm k.k. Notice that for each map u :(M, g) −→ S2, we have

2 0 = ∆g kuk = 2 hu, ∆gui + 4e(u) where

1 ∂u ∂u e(u) = gαβ(x) , , 2 ∂xα ∂xβ 2 ∼ 2 2 and hence, since Nu(x)S = Ru(x) (where Nu(x)S is the normal space of S at u(x)), we have:

⊥ P (∆gu) = hu, ∆gui u u (18) = − 2e(u)u But if u is an B2-valued harmonic map,

Pu(∆gu) = 0 which, together with (18) gives

∆gu + 2e(u)u = 0 (19) This concludes our section on background and prerequisites. Now we are ready to put our knowledge into action and get into the core of the paper.

5 2 Regularity of weakly harmonic maps of surfaces 2.1 Main theorem Here is the main theorem of the paper: Main theorem ([4]). Let N be a C2 compact Riemannian manifold without boundary. Let (M, g) be a Riemannian surface, and let u ∈ H1(M, N ). Then, if u is weakly harmonic, u is C1,α. Furthermore, if the embedding of N into RN is Cl,α for some l ≥ 2, then u is also Cl,α. So, under very mild conditions on N and u, we get an unexpected gain in regularity of u.

2.2 Reduction to the case M = B2 First of all, because smoothness is local, it sufﬁces to show that the above theorem is valid in the neighborhood of each point p ∈ M. Moreover, we have the following two theorems: Theorem 4. Let (M,g) be a Riemannian surface. Then, for each point p ∈ (M,g), there is a neighborhood 2 2 2 Ωp of p in M, and a diffeomorphism Tp from the disk D = (x, y) ∈ R | x + y < 1 to Ωp, such that, if c is the canonical Euclidean metric on D, Tp :(D, c) −→ (Ωp, g) is a conformal map (i.e., it preserves angles).

−1 We also say that Tp is a local conformal chart in (M,g), and that (x, y) are conformal coordinates. For a proof, see [1]. Theorem 5. The set of harmonic functions over an open subset Ω of a Riemannian surface (M,g) depend only on the conformal structure of this surface.

Proof. If (M, g) and (N , h) are two Riemannian surfaces and Ω, V two open subsets of Mand N , then, if T : (Ω, g) −→ (V, h) is a conformal diffeomorphism, we have:

1 E(Ω,g)(φ ◦ T ) = E(V,h)(φ), ∀φ ∈ C (V, R) and

∆g(φ ◦ T ) = λ(∆hφ) ◦ T where

1 ∂T i ∂T j λ = gαβ(x)h (T (x)) 2 ij ∂xα ∂xβ

Notice that, with those two theorems in hand, we can express the Dirichlet integral over Ωp of a map φ : M −→ R in the following simple form: " # Z Z 1 ∂φ ◦ T )2 ∂φ ◦ T 2 e(φ)dvolg = + dxdy 2 Ωp D 2 ∂x ∂y and φ will be harmonic if and only if

∂2(φ ◦ T ) ∂2(φ ◦ T ) ∆(φ ◦ T ) = + = 0 ∂x2 ∂y2 The point is that, when studying harmonic maps of Riemann surfaces, we may suppose, at least locally, that the domain manifold is Euclidean and the domain metric is ﬂat. Hence, to prove our main theorem, it is enough to prove the result for maps u ∈ H1(B2, N ), where B2 is the Euclidean unit ball.

6 2.3 Outline of the proof, difﬁculties, and how they are overcome The work in this subsection is purely formal, but will guide us throughout the main proof. In fact, the goal of the rest of the paper is to overcome all the encountered difﬁculties.

Step 1: Constructing an orthonormal tangent frame field First, we would like to define a smooth orthonormal tangent frame field over (N , h), i.e. for every y ∈ N , an orthonormal basis for TyN . We will see later that this frame field will be crucial to our proof. More precisely, if y ∈ N , define e˜ by:

e˜(y) = (˜e1(y), ..., e˜n(y))

with e˜(y) an orthonormal basis for TyN .

Step 2: Adapting the orthonormal frame We would like to write the equations satisfied by a weakly harmonic map u ∈ H1(B2, N ) by projecting them along the moving frame on N . The advantage of this is an enormous simplification in those equations. Basically, this projection will eliminate most of the non-linearity of the problem, which will lead us later to our smoothness estimates. However, this simplification will not happen for any chosen frame. We must adapt our frame to u, and we will get something called a Coulomb frame. More precisely, define the Gauge group as follows: Definition. 1 2 1 2 H (B ,SO(n)) = R ∈ H (B ,Mn(R)) | R ∈ SO(n)a.e. a 1 2 2 and for every R = [Rb ] ∈ H (B ,SO(n)), define, for a.e. z ∈ B

e(z) = (e1(z), ..., en(z)) where

n X a ea(z) := e˜b(u(z))Rb (z) (20) b=1 Now, among all the R’s in H1(B2,SO(n)), choose an R that minimizes the functional:   Z n 2 2 X ∂ea ∂ea F (R) =  , eb + , eb  dxdy 2 ∂x ∂y B a,b=1 By an easy differentiation, we see that the frame e(z) obtained satisﬁes, in particular the conservation law:

2 div(F) = 0 in B (21) where

∂e ∂e F = a , e , a , e ∂x b ∂y b Deﬁnition. The frame e(z) constructed is called a Coulomb frame D E ∂ea ∂ea Note. An important consequence of (21) is that the coefﬁcients ∂x , eb and ∂y , eb belong to the Lorentz space L(2,1)(B2) instead of just L2(B2).

7 Lorentz spaces For sake of completeness, here is a deﬁnition of a Lorentz space. It may be viewed as a ’twist’ of the Lp spaces.

Deﬁnition. Let f :Ω −→ R be a measurable function. Then the non-increasing arrangement of |f| on [0, |Ω|) is the unique function, denoted by f ∗ : [0, Ω) −→ R which is non-increasing, and such that: µ ({ x ∈ Ω | |f(x)| ≥ s }) = µ ({ t ∈ (0, |Ω|) | f ∗(t) ≥ s) }

Deﬁnition. Let ω be an open subset in RM , p ∈ (1, ∞), q ∈ [1, ∞]. Then the Lorentz space L(p,q)(Ω, R)is the set of measurable functions f :Ω −→ R such that, if we deﬁne: 1 Z t f ∗∗(t) = f ∗(s)ds t 0 and 1 +∞ q Z 1 dt p ∗∗ q kfk(p,q) = (t f (t)) , q < +∞ 0 t and 1 p ∗∗ kfk(p,∞) = sup t f (t) t>0 then

kfk(p,q) < +∞

Step 3: Use the fact that u is weakly harmonic So far, all our results are true even if u is not weakly harmonic, mainly because of equation (21). In this section, the assumption that u is harmonic is be crucial. Deﬁne:

∂u ∂u αa := (z), e (z) − i (z), e (z) ∂x a ∂y a and

∂e i ∂e ωa := b (z), e (z) + b (z), e (z) b ∂x a 2 ∂y a ∂e = b (z), e (z) ∂z¯ a

∂ 1 ∂ ∂ a a (where, as usual, z = x + iy and ∂z¯ = 2 ∂x + i ∂y ). We see that α and ωb satisfy: n ∂αa X = ωaαb (22) ∂z¯ b b=1 From this, and the fact that u is weakly harmonic, it is possible to deduce that u is Locally Lipschitz, by showing that αa is locally bounded in L∞. Because of our particular choice of orthonormal frame, we can b (2,1) 2 show that ωa ∈ L (B ), and this sufﬁces to conclude.

8 Problems and how they will be solved There are many problems that arise in the above outline, the main one being: Does e, as constructed above, exist? And is it possible to minimize F ? It is easy to construct examples of manifolds N where both of those answers are no. However, we will solve the problem by constructing an isometric embedding J of N into another manifold Nˆ , in such a way that if u is weakly harmonic in N , then J ◦ u is weakly harmonic in Nˆ . At the same time, Nˆ will have a frame field e˜ defined over it. Then, we can deform e˜ ◦ u into a Coulomb frame (this is the ’embedding-method’ of constructing Coulomb frames). Alternatively, we can also construct a Coulomb frame adapted to u directly, by approximating u by smooth maps u, and constructing Coulomb frames e associated to the u. We will show that the e will satisfy sufficiently good estimates for us to be able to pass tothe limit when → 0 (this is the ’approximation-method’ of constructing Coulomb frames). So, the gist of the proof is to combine those two methods judiciously in order to construct an adequate Coulomb frame associated to u.

2.4 Reduction to the case where the image manifold is diffeomorphic to a torus Now we can ﬁnally start the ofﬁcial proof of the theorem. We will start with the embedding method described above. For this, we need the following crucial lemma, the proof of which will be omitted:

Lemma 6. Let N be a compact n-dimensional submanifold, without boundary, and Ck embedded in (RN , h., .i), where k ≥ 4. Then there exists an N-dimensional submanifold, Nb, Ck−1 isometrically embedded in (RNb , h., .i) and a Ck−1 embedding, J, of N into Nb, such that:

1. J :(N h., .i N ) −→ (Nb h., .i Nc) is an isometric embedding. R R 2. Nb is diffeomorphic to the torus TN := (S1)N . 3. For any Riemannian manifold (M, g) of dimension m ≥ 1, for any open set Ω of M and for any map u ∈ H1(Ω, N ), if u is weakly harmonic, then so is J ◦ u.

Because of this result, we can now reduce the study of weakly harmonic maps to the case where the image manifold is diffeomorphic to a torus (as long as the image manifold we start with is at least C4). This allows 1 us to construct a C orthonormal frame ﬁeld over the image manifold, denoted by e˜(y) = (˜e1(y), ..., e˜n(y)).

2.5 Construction of a Coulomb frame Let F be the fiber bundle of orthonormal frames over N . Its pull-back by u, denoted by u∗F, is the (not necessarily smooth) bundle over Ω whose fiber over a point x ∈ Ω is the set of orthonormal frames of Tu(x)N . From e˜, which we may interpret as a section of F, we construct a section e˜ ◦ u of u∗F. In fact, it consists of the map x 7−→ (e1(y), ..., en(y)). The following two lemmas then enable us to construct, using e ◦ u, a Coulomb frame over Ω, i.e. a finite energy harmonic section of the fiber bundle u∗F.

Lemma 7. Let N be an n-dimensional compact manifold without boundary, C2 embedded in RN . Let (M, g) 1 be a Riemannian manifold and Ω an open subset of (M, g). Let u ∈ H (Ω, N ) and e¯ = (¯e1, ..., e¯n) be any ∗ ﬁnite energy section of u F. Then, there exists a Coulomb frame over Ω associated to u, e = (e1, ..., en), i.e. a ﬁnite energy section of u∗F such that: ( ∂ gαβ(x)pdet(g(x)) ∂ea , e = 0 in Ω ∂xα ∂xβ b (23) α ∂ea n (x) ∂xα , eb = 0 on ∂Ω where n is the normal vector on ∂Ω. Moreover, we have the estimates:

T 2 T 2 e de L2(Ω) ≤ e¯ de¯ L2(Ω) (24) and

2 T 2 2 kdekL2(Ω) ≤ e de L2(Ω) + C kdukL2(Ω) (25)

9 where C is a constant that depends only on N . In the case e¯ =e ˜ ◦ u, where e˜ is a C1 section of F, we deduce from (24) and (25) that

2 2 kdekL2(Ω) ≤ C kdukL2(Ω) (26)

Note. We can vie w e in essentially two ways. On the one hand, e can be viewed as a harmonic section of u∗F. On the other hand, and this point of view is somewhat more convenient, one can consider e as a map in H1(Ω, (RN )n), which, to almost all x ∈ Ω associates a family of n orthonormal vectors in RN that form a basis for Tu(x)N . So e can be represented as a n × N matrix! More precisely:

 1 1  e1 ... en  . .  e = (e1, ..., en) =  . .  N N e1 ... en Hence, eT de should be interpreted as a matrix product; in this case it will be the n × n matrix whose 1 ∗ elements are the 1-forms hdea, debi. Likewise, the action of the gauge group H (Ω,SO(n)) on u F will be represented by the matrix product

e 7−→ eR for any R ∈ H1(Ω,SO(n)), where

b ea 7−→ ebRa Proof. We will proceed in several steps

Step 0: Preliminaries First of all, if R ∈ H1(Ω,SO(n)), then deﬁne the moving frame e by:

a ea(x) =e ¯b(x)Rb (x) and deﬁne Z 1 T 2 F (R) = e de dvolg 2 Ω where n T 2 αβ X ∂ea ∂ea e de = g (x) , eb , eb ∂xα ∂xβ a,b=1

Step 1: Existence of a Minimum for F

1 Let (Rk)k∈N be a sequence in H (Ω,SO(n)) minimizing F. Deﬁne

ek(x) =e ¯(x)Rk(x) 1 First, let’s show that (Rk) is bounded in H (Ω,SO(n)). By expanding, we see that:

T 2 T 2 T T T 2 e dek = e¯ de¯ + 2 e¯ de,¯ dRkR + dRkR k k k (27) T 2 ≥( e¯ de¯ − |dRk|)

T 2 2 T 2 However, e¯ de¯ is bounded in L (Ω), and, since (Rk) minimizes F, ek dek is also bounded in L (Ω). 2 Hence, by (27), |dRk| is also bounded in L (Ω). Hence, we can extract a subsequence of k (which, we’ll still call k) such that there exists R ∈ H1(Ω,SO(n)) such that

10 2 dRk * dR weakly in L (Ω,Mn(R)) 2 (28) Rk → R in L (Ω,SO(n)) and

Rk → R a.e. (29)

T T 2 Using the dominated convergence theorem, we deduce from (29) that e¯ deR¯ k − e¯ deR¯ goes to 0 in L1(Ω), and hence:

T T 2 e¯ deR¯ k → e¯ deR¯ in L (Ω,Mn(R)) (30) T T T But, by deﬁnition of the transpose, e¯ de,¯ dRkRk = e¯ deR¯ k, dRk . Hence, using this and the con- vergences (28) and (30), we get that: Z Z T T T T lim e¯ de,¯ dRkRk dvolg = e¯ de,¯ dRR dvolg (31) k→∞ Ω Ω Similarly, we deduce from (28) that: Z Z T T lim dRkRk dvolg ≥ dRR dvolg (32) k→∞ Ω Ω Now, putting together (27), (31), and (32), we conclude that:

lim F (Rk) ≥ F (R) (33) k→∞

Hence, R minimizes F . We denote by e = (e1, ..., en) the frame associated to R.

Step 2: Euler-Lagrange equation Given a test function φ ∈ C∞(Ω, so(n)), and for close to zero, deﬁne

R =R(x)exp(φ(x)) =R(x)(1 + φ(x) + o()) and

e(x) =¯e(x)R(x) =e(x)(1 + φ(x) + o()) Since R minimizes F, and hence is a ’critical point’ of F, we have

F (R) = F (R) + o() And a calculation yields

T T T e de = e de + (dφ − φe de) + o() Hence, we get: Z T T e de, dφ − φe de dvolg = 0 (34) Ω However, because φ ∈ so(n), eT de, φeT de always vanishes, which simpliﬁes (34), and, by integrating by parts, we get: Z Z T ∂e T φ, e dσ − φ, divg(e de) dvolg = 0 (35) ∂Ω ∂n Ω

11 where

1 ∂ ∂e div (eT de) = √ gαβ(x)pdetg eT g detg ∂xα ∂xβ And from this, we easily get that (35) implies (23).

Estimates (24) and (25) Estimate (24) is a consequence of the construction of e, obtained by minimization. As for (25), for any N ⊥ y ∈ N , denote by P (y) the orthogonal projection of R onto TyN , and P (y) = 1N − P (y). It follows from the hypothesis on N that P and P ⊥ are C1-maps on N . Using the identity

⊥ dei = P (u)(dei) + P (u)(dei) we deduce

n n 2 X 2 X ⊥ 2 kdekL2(Ω) = kP (u)(dea)kL2(Ω) + P (u)(dea) L2(Ω) a=1 a=1 n (36) T 2 X ⊥ 2 = e de L2(Ω) + P (u)(dea) L2(Ω) a=1 ⊥ However, because d(P (u)(ea) = 0, we obtain:

n n X ⊥ 2 X ⊥ 2 P (u)(dea) L2(Ω) = dP (u)(ea) L2(Ω) a=1 a=1 (37) 2 2 ≤ kP kW 1,∞ kdukL2(Ω) and (36) and (37) together imply that

2 T 2 2 2 kdekL2(Ω) ≤ e de L2(Ω) + kP kW 1,∞ kdukL2(Ω) which is (25)!

The following lemma provides a somewhat more direct way of constructing a ﬁnite energy Coulomb orthonormal frame for u (it is an illustration of the ’approximation’-method for constructing Coulomb frames announced in the plan of the proof). It resembles lemma 7, but has hypotheses that are somewhat easier to check. Furthermore, after having proven this lemma, we won’t need lemma 7 any more.

Lemma 8. Let N be a compact submanifold without boundary, C2 embedded in RN . Let B2 be the unit ball 2 1 2 in R and u a map in H (B , N ). Then there exists a constant γ0 depending only on N such that if

2 kduk 2 2 ≤ γ0 (38) L (B ) then we can construct a ﬁnite energy Coulomb orthonormal frame for u (i.e. a harmonic section of u∗F).

Proof-Sketch. Step 1

We start by working with a map u ∈ C2(B2, N ) ∩ H1(B2, N ) satisfying (38). Since u is smooth and B2 is contractible, it is possible to construct a section e¯ of u∗F (where F is the bundle of orthonormal tangent frames on N ) belonging to C1(B2) ∩ H1(B2). We will work with e¯ as a map in H1(B2,M(n × N, R)).

12 Step 2

1 2 1 T 2 Let e(z) =e ¯(z)R(z), where R ∈ H ( ,SO(n)), and we minimize F (R) = e de 2 2 just like we B 2 L (B ) did in lemma 7. In this way, we obtain a Coulomb frame which we still denote by e(z) = (e1(z), ..., en(z)). a 1 2 Thanks to equation (23), there exist maps in Ab ∈ H (B ) such that: ∂Ab ∂Ab ∂e ∂e a , a = a , e , − a , e (39) ∂x ∂y ∂y b ∂x b A direct calculation then yields

N b X k k 2 −∆Aa = ea, eb in B (40) k=1 where, in general ∂a ∂b ∂a ∂b {a, b} := − ∂x ∂y ∂y ∂x And the boundary condition in (23) implies that we may choose

b 2 Aa = 0 on ∂B (41)

Using Wente’s inequality, we deduce from (40) and (41) that there exists a universal constant C1 such that

n 2 X 2 4 kdAk 2 2 := kdAik 2 2 ≤ C1 kdek 2 2 (42) L (B ) L (B ) L (B ) i,j=1

Step 3 Now, putting together inequality (42) with inequality (25) of lemma 7, i.e.

2 2 2 2 kdek 2 2 ≤ kdAk 2 2 + C kduk 2 2 L (B ) L (B ) L (B ) we get

2 2 C1t − t + C2 kduk 2 2 ≥ 0 (43) L (B ) 2 where t = kdek 2 2 . Note that if L (B )

2 −1 kduk 2 2 < γ0 := (4C1C2) L (B ) −1 then the polynomial in (43) would take negative values in a neighborhood of t = (2C1) , and hence 2 kdek 2 2 cannot approach this value! L (B )

Step 4

2 We go over the previous steps once again, but this time over balls of variable size Br, 0 < r < 1. One can 2 2 show that kderk 2 2 is a continuous function of r, and so, if (38) is true, then kderk 2 2 cannot approach L (Br ) L (Br ) −1 t = (2C1) . By a continuity argument, we deduce that

2 −1 kdek 2 2 ≤ (2C1) L (B )

13 Step 5

2 2 1 2 We use the fact that C (B , N ) is dense in H (B , N ), and applying the previous steps to a sequence uk in C2(B2, N ) approaching u in H1(B2, N ), we can construct a sequence of bounded energy harmonic sections ∗ of ukF. Passing to the limit, we obtain our result.

2.6 Gain of regularity The following lemma speciﬁes exactly what gain of regularity one obtains by using a Coulomb frame in dimension 2. It is crucial to prove our main theorem.

Lemma 9. Let N be a compact Riemannian manifold without boundary, C2 embedded in RN . Let u ∈ H1(B2, N ) and e be a Coulomb frame, i.e. a section of u∗F belonging to H1(B2)and satisfying (23). Then all the coefﬁcients of eT de belong to L(2,1)(B2). Moreover, there exists a constant C such that:

T 2 4 e de (2,1) 2 ≤ kdek 2 2 (44) L (B ) L (B ) Furthermore, in the case where the frame e is obtained by minimization, the the gauge orbit of a frame e¯ =e ˜ ◦ u (see lemma 7), we have the estimate

T 2 4 e de (2,1) 2 ≤ kduk 2 2 (45) L (B ) L (B )

Proof. Inequality (45) follows from (44) and of inequality (26) in lemma 7. In order to show (44), recall the b 1 2 b quantities Aa ∈ H (B ) deﬁned by (39). Then, using equations (40) and (41) satisﬁed by Aa, and applying b (2,1) 2 theorem 10 (see below), we obtain that Aa ∈ L (B ), and

b 2 2 2 dA 2 ≤ C kdeak 2 2 kdebk 2 2 (46) a L(2,1)(BB ) L (B ) L (B ) which yields (44) because of (39). Here is the theorem (whose proof is omitted) we need in order to complete the above proof:

Theorem 10. Let Ω be an open subset of R2, with C1-boundary, a, b ∈ H1(Ω), and let φ be a solution of: −∆φ = {a, b} := ∂a ∂b − ∂a ∂b in Ω ∂x ∂y ∂y ∂x (47) φ = 0 on ∂Ω

∂φ ∂φ (2,1) Then ∂x , ∂y ∈ L (Ω), and

kdφkL(2,1)(Ω) ≤ C(Ω) kdakL2(Ω) kdbkL2(Ω) (48) where C(Ω) is a constant depending only on Ω.

2.7 At long last: Proof of the main theorem

Let u ∈ H1(B2, N ) be a weakly harmonic map. Assume (by placing ourselves in a sufﬁciently small ball) 2 that kduk 2 2 ≤ γ0. Now, by lemma 8, we obtain a ﬁnite energy Coulomb frame e for u. Moreover, by L (B ) T (2,1) 2 a a lemma 9, we know that e de ∈ L (B ). Now, let α and ωb like in the outline of the proof. Now write:

n X du = hdu, eai ea (49) a=1 and using the relations:

( du = 0 ∂u ∂u (50) d ∂x dy − ∂y dx = ∆udx ∧ dy⊥TuN

14 we get that:

n ∂αa X = ωaαb (51) ∂z¯ b b=1 where the real part of (51) corresponds to the ﬁrst part of (50), and similar for the imaginary part. Now, a a deﬁning α to be the column vector of the α s, and ω ∈ SO(n) ⊗ C to be the matrix of the ωb (notice that the diagonal entries of ω are all 0), we can conveniently rewrite (51) as: ∂α = ωα (52) ∂z¯ The rest of the proof boils down to studying equation (52).

∞ 2.7.1 Existence of solutions of (52) in Lloc

2 2 n 1 T ∂e T ∂e First of all, notice that α ∈ L (B , C ), and that, because of lemma 9, and the fact that ω = 2 e ∂x + ie ∂y , ω ∈ L(2,1)(B2, so(n) ⊗ C). z ∈ 2 z 2 2 Now, let 0 B , and choose a ball centered at 0 and contained in B , denoted by Bz0 , such that: 1 ω 2 < √ Bz0 L(2,1) 3 2π β ∈ L∞( 2 , GL(n, )) We would like to construct Bz0 C a solution of ∂β = ωβ in 2 (53) ∂z¯ Bz0 ω 2 ∼ ω 2 2 We temporarily denote by the function on R = C that coincides with in Bz0 , and is 0 outside Bz0 . 1 Notice that we’ll always have ω 2 < √ . Bz0 L(2,1) 2 2π Now deﬁne the linear operator T : L∞(C,M(n, C)) → L∞(C,M(n, C)) by: Z ω(v)β(v) 1 2 1 T (β)(z) = dv dv = Lω ◦ ∗ β(z) (54) V π(z − v) πz

where ∗ denotes convolution, and Lω left-multiplication by (the matrix) ω. (2,1) 2 ∗ (2,∞) 2 1 (2,∞) 2 Now, because L (R ) = L (R ) (see [10]), and because πz ∈ L (R ), we get that T is continuous, with

1 1 kT kL∞ ≤ kωkL(2,1) < (55) πz L(2,∞) 3 Applying T to both sides of (53), we see that any solution β of (53) should be a solution of

β − T (β) = H (56)

where H : C −→ M(n, C) is holomorphic. Using estimate (55), we conclude, using a ﬁxed point argument, that if H ∈ L∞(C), (56) has a unique solution β˜. Choosing in particular H ≡ 1, the ’constantly- 1’-function, we get that 2 β˜ − 1 ≤ L∞ 3 and that β˜ takes values in GL(n, C). Furthermore, β˜ is a solution of (53). β 2 To reiterate, we had that any solution of (53) restricted to Bz0 is a solution of (56). However, (56) has a β˜ 2 β˜ unique solution , from which it follows that (53) has a unique solution in Bz0 , namely . From now, on, we denote by β the solution β˜ just constructed. 2 β Finally, we use over Bz0 , the solution , and deduce from (52) that

15 ∂ (β−1α) = (β−1(ω − ω)α) = 0 ∂z¯ β−1α 2 α ∈ L∞ ( 2, n) Hence is holomorphic in Bz0 , and, in particular, locally bounded. Hence, loc B C , and ∞ hence u is locally Lipschitz. Now from (16), we get that ∆(u) ∈ Lloc, and hence, by the Sobolev embedding theorems (see [2]), this implies that u ∈ C1,α. Finally, all higher regularity follows from the following theorem (the proof of which is being omitted, but see [6]): Theorem 11. Every continuous weakly harmonic map u ∈ H1(M, N ) is smooth. More precisely, if g (the metric on M) is Ck,α, and h (the metric on N ) is Cl,α, then u is Cmin(k+1,l+1)α.

This concludes our proof of the main theorem.

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