First boundary Dirac eigenvalue and boundary capacity potential

TL;DR

This paper establishes new lower bounds for the first Dirac eigenvalue on hypersurfaces in spin asymptotically flat manifolds, linking spectral properties with boundary capacity and geometric inequalities.

Contribution

It introduces novel lower bounds involving boundary capacity potential and capacity, and develops estimates for Dirac eigenvalues with singular metrics on compact manifolds.

Findings

Derived lower bounds for Dirac eigenvalues involving boundary capacity

Established geometric inequalities connecting spectral data and boundary capacity

Provided estimates for Dirac eigenvalues with singular metrics

Abstract

We derive new lower bounds for the first eigenvalue of the Dirac operator of an oriented hypersurface $\Sigma$ bounding a noncompact domain in a spin asymptotically flat manifold (M n , g) with nonnegative scalar curvature. These bounds involve the boundary capacity potential and, in some cases, the capacity of $\Sigma$ in (M n , g) yielding several new geometric inequalities. The proof of our main result relies on an estimate for the first eigenvalue of the Dirac operator of boundaries of compact Riemannian spin manifolds endowed with a singular metric which may have independent interest.