Boundary value problems for noncompact boundaries of Spin^cmanifolds and spectral estimates

Abstract

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 Ch. Bär 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, and the limiting case is studied.

A generalized MIT Bag operator on spin manifolds in the non-relativistic limit

“…By the definition of the Sobolev spaces, one can observe that it is possible to extend the results valid for Euclidean spaces. We state a trace theorem which is a modification of [8,Theorem 3.7], where we add a bound for the L 2 -norm of the trace.…”

“…The computations for the forms of square operators is done in section 4 and the main tool used to this aim is the Schrödinger-Lichnerowicz formula, which gives the expression of the square of the Dirac operator on a spin manifold. Once we get the sesquilinear forms, the graph norm of A m is shown to be equivalent to the H 1 norm on its domain, and we can use the analysis done in [8] to conclude on self-adjointness.…”

The index of Callias-type operators with Atiyah–Patodi–Singer boundary conditions

“…They obtained the Fredholm property for Calliastype operators with APS boundary conditions, making it possible to study the index problem on noncompact manifolds with boundary. The results in [6] were also partially generalized to Spin c manifolds of bounded geometry with noncompact boundary in [22].…”

Sobolev spaces on Riemannian manifolds with bounded geometry: General coordinates and traces

“…In Section , we establish analogous results for vector bundles of bounded geometry. An application of our trace result for vector bundles, Theorem , may be found in , where the authors classify boundary value problems of the Dirac operator on bundles of bounded geometry, deal with the existence of a solution, and obtain some spectral estimates for the Dirac operator on hypersurfaces of bounded geometry.…”