Dirac eigenvalues and the hyperspherical radius

TL;DR

This paper establishes an upper bound for the smallest Dirac eigenvalue on closed Riemannian spin manifolds using the hyperspherical radius, leading to various geometric implications and new estimates for special classes of manifolds.

Contribution

It provides a novel upper estimate for Dirac eigenvalues in terms of the hyperspherical radius and explores its geometric consequences, including applications to Kähler and quaternionic Kähler manifolds.

Findings

Upper bound for Dirac eigenvalues in terms of hyperspherical radius

Connections to scalar curvature rigidity and Yamabe constant comparisons

Improved estimates for special classes of manifolds such as Kähler and quaternionic Kähler

Abstract

For closed connected Riemannian spin manifolds an upper estimate of the smallest eigenvalue of the Dirac operator in terms of the hyperspherical radius is proved. When combined with known lower Dirac eigenvalue estimates, this has a number of geometric consequences. Some are known and include Llarull's scalar curvature rigidity of the standard metric on the sphere, Geroch's conjecture on the impossibility of positive scalar curvature on tori and a mean curvature estimate for spin fill-ins with nonnegative scalar curvature due to Gromov, including its rigidity statement recently proved by Cecchini, Hirsch and Zeidler. New applications provide a comparison of the hyperspherical radius with the Yamabe constant and improved estimates of the hyperspherical radius for K\"ahler manifolds, K\"ahler-Einstein manifolds, quaternionic K\"ahler manifolds and manifolds with a harmonic 1-form of constant length.