An elliptic theory of indicial weights and applications to non-linear geometry problems

TL;DR

This paper develops an elliptic theory for indicial weights on non-compact manifolds, extending Fredholm properties even at indicial weights, with applications to Yang-Mills theory and minimal surfaces.

Contribution

It introduces a new elliptic theory that applies at indicial weights, broadening the understanding of elliptic operators on non-compact manifolds.

Findings

Elliptic theory extends to indicial weights.

Fredholm property holds at indicial weights.

Applications demonstrated in Yang-Mills and minimal surface problems.

Abstract

Given an elliptic operator $P$ on a non-compact manifold (with proper asymptotic conditions), there is a discrete set of numbers called indicial roots. It's known that $P$ is Fredholm between weighted Sobolev spaces if and only if the weight is not indicial. We show that an elliptic theory exists even when the weight is indicial. We also discuss some simple applications to Yang-Mills theory and minimal surfaces.