Uniqueness in Rough Almost Complex Structures and Differential Inequalities

TL;DR

The paper proves a uniqueness property for J-holomorphic discs in almost complex structures of class at least 1/2, contrasting with known non-uniqueness results, and explores related differential inequalities.

Contribution

It establishes a new uniqueness theorem for J-holomorphic discs with minimal regularity and investigates related differential inequalities in almost complex manifolds.

Findings

J-holomorphic discs constant on open sets are globally constant under certain regularity.

Differential inequalities influence uniqueness properties in almost complex structures.

Vector and scalar cases exhibit different behaviors in uniqueness questions.

Abstract

We prove that for almost complex structures of H\"older class at least 1/2, any J-holomorphic disc, that is constant on some non empty open set, is constant. This is in striking contrast with well known, trivial, non-uniqueness results. We also investigate uniqueness questions (do vanishing on some open set, or vanishing to infinite order, or having a non isolated zero, imply vanishing) in connection with differential inequalities that arise in the theory of almost complex manifolds. The case of vector valued functions is different from the case of scalar valued functions.