Dirac Cohomology on Manifolds with Boundary and Spectral Lower Bounds

TL;DR

This paper extends Hodge theory to Dirac bundles on manifolds with boundary, providing spectral lower bounds and applications to the Atiyah-Singer operator, including eigenvalue estimates for spin manifolds.

Contribution

It generalizes Cheeger's spectral lower bound technique to Dirac operators and develops a method to estimate Dirac spectra using submanifold covers.

Findings

Established a Dirac Hodge decomposition with boundary conditions.

Generalized spectral lower bounds for Dirac Laplacians.

Proved eigenvalue bounds for spin manifolds under degenerating geometries.

Abstract

Along the lines of the classic Hodge-De Rham theory a general decomposition theorem for sections of a Dirac bundle over a compact Riemannian manifold is proved by extending concepts as exterior derivative and coderivative as well as as elliptic absolute and relative boundary conditions for both Dirac and Dirac Laplacian operators. Dirac sections are shown to be a direct sum of harmonic, exact and coexact spinors satisfying alternatively absolute and relative boundary conditions. Cheeger's estimation technique for spectral lower bounds of the Laplacian on differential forms is generalized to the Dirac Laplacian. A general method allowing to estimate Dirac spectral lower bounds for the Dirac spectrum of a compact Riemannian manifold in terms of the Dirac eigenvalues for a cover of 0-codimensional submanifolds is developed. Two applications are provided for the Atiyah-Singer operator. First, we prove the existence on compact connected spin manifolds of Riemannian metrics of unit volume with arbitrarily large first non zero eigenvalue, which is an already known result. Second, we prove that on a degenerating sequence of oriented, hyperbolic, three spin manifolds for any choice of the spin structures the first positive non zero eigenvalue is bounded from below by a positive uniform constant, which improves an already known result.