The Linearized Floer Equation in a Chart

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.