Boundary value problems for noncompact boundaries of Spin$^c$ manifolds and spectral estimates

TL;DR

This paper extends boundary value problem theory for Dirac operators on noncompact Spin$^c$ manifolds with bounded geometry, deriving spectral bounds involving boundary mean curvature and exploring limiting cases with examples.

Contribution

It generalizes existing boundary value problem theory to noncompact boundaries of Spin$^c$ manifolds and establishes spectral estimates involving boundary mean curvature.

Findings

Derived lower bounds for Dirac spectrum involving boundary mean curvature

Extended boundary value problem theory to noncompact Spin$^c$ manifolds

Analyzed limiting cases and provided examples

Abstract

We study boundary value problems for the Dirac operator on Riemannian Spin$^c$ manifolds of bounded geometry and with noncompact boundary. This generalizes a part of the theory of boundary value problems by C. B\"ar and W. Ballmann for complete manifolds with closed boundary. As an application, we derive the lower bound of Hijazi-Montiel-Zhang, involving the mean curvature of the boundary, for the spectrum of the Dirac operator on the noncompact boundary of a Spin$^c$ manifold. The limiting case is then studied and examples are then given.