The Linearized Floer Equation in a Chart

TL;DR

This paper introduces a new mathematical structure called almost extendable weak Hessian field, analyzing the Hessian of the area functional in non-Darboux charts and proving a Fredholm theorem for related operators.

Contribution

It presents the first study of the Hessian of the area functional in non-Darboux charts and introduces the almost extendable weak Hessian field structure.

Findings

Established a Fredholm theorem for Robbin-Salamon operators

Identified a new mathematical structure for non-Darboux Hessians

Analyzed the Hessian of the area functional in a novel setting

Abstract

In this article, we are considering the Hessian of the area functional in a non-Darboux chart. This does not seem to have been considered before and leads to an interesting new mathematical structure which we introduce in this article and refer to as almost extendable weak Hessian field. Our main result is a Fredholm theorem for Robbin-Salamon operatorsassociated to non-continuous Hessians which we prove by taking advantage of this new structure.