A remark on a theorem of Schoen and Wolfson

Abstract

Let $l:\Sigma \to X$ be a weakly Lagrangian map of a compact orientable surface $\Sigma$ in a Kähler surface $X$ which is area minimizing in its homotopy class of maps in $W^{1,2}(\Sigma ,X)$, the Sobolev space of maps of square integrable first derivative. Schoen and Wolfson showed such $l$ is Lipschitz, and it is smooth except at most at finitely many points of Maslov index 1 or -1. In this note, we observe if in addition c_1(X)[l]=0, $l$ is smooth everywhere. Here $c_1(X)$ is the first Chern class of $X$.

This paper is incorrect. It is withdrawn by the author