A remark on a theorem of Schoen and Wolfson

Abstract

Let $ l : Σ\to X$ be a weakly Lagrangian map of a compact orientable surface $ Σ$ in a Kähler surface $ X$ which is area minimizing in its homotopy class of maps in $ W^{1,2}(Σ, 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$.