Elliptic boundary value problem on non-compact $G$-manifolds

TL;DR

This paper establishes a generalized index equality for elliptic boundary value problems on non-compact G-manifolds, extending previous results to non-compact group actions and manifolds.

Contribution

It proves an index equality between Hochs-Mathai and Atiyah-Patodi-Singer types on non-compact manifolds with non-compact group actions, broadening the scope of prior compact cases.

Findings

Proves index equality for non-compact G-manifolds.

Analyzes Fredholm properties of elliptic boundary value problems.

Extends previous compact manifold results to non-compact settings.

Abstract

In this paper, an equality between the Hochs-Mathai type index and the Atiyah-Patodi-Singer type index is established when the manifold and the group action are both non-compact, which generalizes a result of Ma and Zhang for compact group actions. As a technical preparation, a problem concerning the Fredholm property of the global elliptic boundary value problems of the Atiyah-Patodi-Singer type on a non-compact manifold is studied.