Boundary Problems for Dirac-Type Operators on Manifolds with Multi-Cylindrical End Boundaries

Abstract

Abstract. The goal of this paper is to establish a geometric program to study elliptic pseudodifferential boundary problems which arise naturally under cutting and pasting of geometric and spectral invariants of Dirac type operators on manifolds with corners endowed with multi-cylindrical, or b-type, metrics and 'b-admissible' partitioning hypersurfaces. We show that the Cauchy data space of a Dirac operator on such a manifold is Lagrangian for the self-adjoint case, the corresponding Calderón projector is a b-pseudodifferential operator of order 0, characterize Fredholmness, prove relative index formulae, and solve the Bojarski conjecture.

Geometric BVPs, Hardy spaces, and the Cauchy integral and transform on regions with corners

“…so it can be applied to a wide range of situations. (In [68] the following result was stated for topological vector spaces but the proof is purely linear algebraic and makes no mention of topology.) Theorem 8.1.…”

“…This theorem was proved with J. Park in [68], but in order to keep this article self-contained, we give an abbreviated proof of the theorem whose details can be filled in or looked up in [68]. Let V 0 , V 1 , V 2 be vector spaces (finite-or infinite-dimensional) and let…”

Elliptic boundary problems on manifolds with polycylindrical ends

“…Loya and Park studied in [21] Diractype operators on such noncompact configurations, and they call them "manifolds with multicylindrical end boundaries" (they require an additional but essentially unnecessary topological condition for the hypersurface where the boundary condition is imposed).…”

“…Our focus in this paper is the investigation of general Shapiro-Lopatinsky elliptic boundary value problems on manifolds with polycylindrical ends. Loya and Park studied in [23] Dirac-type operators on such noncompact configurations, and they call them "manifolds with multi-cylindrical end boundaries" (they require an additional but essentially unnecessary topological condition for the hypersurface where the boundary condition is imposed).…”

“…At the hypersurfaces on which the operators are regular elliptic we are imposing boundary conditions. Motivated by questions arising in cutting and pasting in b-geometry, Loya and Park investigated in [23] the Cauchy data space and the Calderón projector at regular hypersurfaces for totally characteristic operators of Dirac-type on manifolds with corners. As is the case also in the classical setting of smooth compact manifolds, their work is not covered by our framework unless the boundary condition for the Dirac operator under consideration is Shapiro-Lopatinsky elliptic (e.g., if it is local).…”