Eigenvalue bounds for the Dirac operator

Abstract

A natural question in the study of geometric operators is that of how much information is needed to estimate the eigenvalues of an operator. For the square of the Dirac operator, such a question has at least peripheral physical import. When coupled to gauge fields, the lowest eigenvalue is related to chiral symmetry breaking. In the pure metric case, lower eigenvalue estimates may help to give a sharper estimate of the ADM mass of an asymptotically flat spacetime with black holes. We use three tools to estimate the eigenvalues of the square of the (purely metric) Dirac operator the conformal covariance of the operator, a patching method and a heat kernel bound.