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

TL;DR

This paper proves that continuous functions satisfying a specific differential inequality have zero sets forming complex varieties and establishes a removable singularity theorem for J-holomorphic maps across polar sets.

Contribution

It introduces new results connecting differential inequalities to complex varieties and extends removable singularity theorems for J-holomorphic maps to polar sets.

Findings

Zero set of functions satisfying |ar{d}f| extless{}|f| is a complex variety.

Continuous J-holomorphic maps extend across polar singularities.

Removable singularity theorem for J-holomorphic maps on polar sets.

Abstract

We consider a continuous function $f$ on a domain in $\mathbf C^n$ satisfying the inequality that $|\bar \partial f|\leq |f|$ 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 $\Omega$.