Differential inequalities of continuous functions and removing singularities of Rado type for J-holomorphic maps

Abstract

We consider a continuous function $f$ on a domain in ${\mathbf{C}}^n$ satisfying the inequality that $|\bar{\partial }f|\le |f|$ off its zero set. The main conclusion is that the zero set of $f$ is a complex variety. We also obtain removable singularity theorem of Rado type for J-holomorphic maps. Let $\Omega$ be an open subset in $\mathbf{C}$ and let $E$ be a closed polar subset of $\Omega$. Let $u$ be a continuous map from $\Omega$ into an almost complex manifold $(M,J)$ with $J$ of class $C^1$. We show that if $u$ is J-holomorphic on $\Omega \setminus E$ then it is J-holomorphic on $\Omega$.