The regularity of conformal target harmonic maps

Abstract

We establish that any weakly conformal $W^{1,2}$ map from a Riemann surface S into a closed oriented sub-manifold $N^n$ of an euclidian space ${\mathbb {R}}^m$ realizes, for almost every sub-domain, a stationary varifold if and only if it is a smooth conformal harmonic map form S into $N^n$.

A Appendix

Proposition A. 1

Let $N^n$ be a $C^2$ sub-manifold of the euclidian space ${\mathbb {R}}^m$. Let $(\Sigma ,h)$ be a compact Riemann surface (equipped with a metric compatible with the complex structure) possibly with boundary. Let $\vec {\Phi }$ be a map in $W^{1,2}(\Sigma ,N^n)$. Assume $\vec {\Phi }$ is weakly conformal and continuous on $\partial \Sigma$. The integer rectifiable varifold associated to $(\vec {\Phi },\Sigma )$ is stationary in $N^n{\setminus }\vec {\Phi }(\partial \Sigma )$ if and only if

$$\begin{aligned} \forall \ {F}\in C^\infty _0(N^n{\setminus }\vec {\Phi }(\partial \Sigma ),{\mathbb {R}}^m)\quad \int _{\Sigma } \left[\left<d(F(\vec {\Phi })), d\vec {\Phi }\right>_h\!-\! F(\vec {\Phi })\ A(\vec {\Phi })(d\vec {\Phi },d\vec {\Phi })_h\right]\ dvol_h\!=\!0 \end{aligned}$$

where $A(\vec {q})(\vec {X},\vec {Y})$ denotes the second fundamental form of $N^n$ at the point $\vec {q}$ and acting on the pair of vectors $(\vec {X},\vec {Y})$ and by an abuse of notation we write

$$\begin{aligned} \ A(\vec {\Phi })(d\vec {\Phi },d\vec {\Phi })_h:=\sum _{i,j=1}^2h_{ij}\ A(\vec {\Phi })(\partial _{x_i}\vec {\Phi },\partial _{x_j}\vec {\Phi }). \end{aligned}$$

$\square$

Proof of proposition A. 1

For any $\vec {q}\in N^n$ one denotes by $P_T(\vec {q})$ the symmetric matrix giving the orthogonal projection onto $T_{\vec {q}}N^n$. The integer rectifiable varifold given by $(\vec {\Phi },\Sigma )$ is by defintion the following Radon measure on $G_2(T{\mathbb {R}}^m)$ the Grassman bundle of un-oriented 2-planes over ${\mathbb {R}}^m$ given by

$$\begin{aligned} \forall \ \phi \in C^\infty (G_2(T{\mathbb {R}}^m))\quad \quad {\mathbf v}_{\vec {\Phi }}(\phi )= & {} \int _{G_2(T{\mathbb {R}}^m)}\phi (S,\vec {q})\ dV_{\vec {\Phi }}(S,\vec {q})\\:= & {} \int _{\Sigma } \phi (\vec {\Phi }_*(T_{x}\Sigma ),{\vec {\Phi }(x)})\ dvol_{g_{\vec {\Phi }}} \end{aligned}$$

By definition (see [1]), the varifold ${\mathbf v}_{\vec {\Phi }}$ is stationary in $N^n$ if

$$\begin{aligned} \forall \ {F}\in C^\infty _0(N^n{\setminus }\vec {\Phi }(\partial \Sigma ),{\mathbb {R}}^m)\quad \int _{\Sigma } \text{ div }_S(P_T\, F)(\vec {q})\ dV_{\vec {\Phi }}(S,\vec {q})=0 \end{aligned}$$

(A. 1)

In local conformal coordinates at a point where $|\partial _{x_1}\vec {\Phi }|=|\partial _{x_2}\vec {\Phi }|=e^\lambda$, introducing the orthonormal basis of $S:=\vec {\Phi }_*T_x\Sigma$ given by $\vec {e}_i:=e^{-\lambda }\partial _{x_i}\vec {\Phi }$, one has by definition

$$\begin{aligned} \text{ div }_{\vec {\Phi }_*T_x\Sigma }(P_T\, F)(\vec {\Phi }):=\sum _{i=1}^2\partial _{\vec {e}_i}(P_T\, F)(\vec {\Phi })\cdot \vec {e}_i=\sum _{i=1}^2\sum _{k=1}^me_i^k\ \partial _{z_k}(P_T\, F)(\vec {\Phi })\cdot \vec {e}_i \end{aligned}$$

where $\vec {e}_i:=\sum _{k=1}^me_i^k\ \partial _{z_k}$. Hence we have

$$\begin{aligned} \displaystyle \text{ div }_{\vec {\Phi }_*T_x\Sigma }(P_T\, F)(\vec {\Phi })= & {} e^{-2\lambda }\,\sum _{i=1}^2\sum _{k=1}^m\partial _{x_i}\Phi ^k\ \partial _{z_k}(P_T\, F)(\vec {\Phi })\cdot \partial _{x_i}\vec {\Phi }\ \\ \displaystyle \quad= & {} e^{-2\lambda }\,\nabla (F(\vec {\Phi }))\cdot \nabla \vec {\Phi }- F(\vec {\Phi })\ \sum _{i=1}^2 e^{-2\lambda }\ A(\vec {\Phi })(\partial _{x_i}\vec {\Phi },\partial _{x_i}\vec {\Phi }) \end{aligned}$$

where we used respectively that $P_T(\vec {\Phi })\nabla \vec {\Phi }=\nabla \vec {\Phi }$ and that $A(\vec {q})(\vec {X},\vec {Y})=-\,\partial _{\vec {X}}P_T(\vec {q})\cdot \vec {Y}$. Multiplying by $dvol_{g_{\vec {\Phi }}}= e^{2\lambda }\ dx_1\wedge dx_2$ we obtain at almost every point x where $\nabla \vec {\Phi }(x)\ne 0$

$$\begin{aligned} \text{ div }_{\vec {\Phi }_*T_x\Sigma }(P_T\, F)(\vec {\Phi })\ dvol_{g_{\vec {\Phi }}}= \left[\left<d(F(\vec {\Phi })), d\vec {\Phi }\right>_{g_{\vec {\Phi }}}- F(\vec {\Phi })\ A(\vec {\Phi })(d\vec {\Phi },d\vec {\Phi })_{g_{\vec {\Phi }}}\right]\ dvol_{g_{\vec {\Phi }}}\nonumber \\ \end{aligned}$$

(A. 2)

This concludes the proof of the lemma. $\square$