Geometric and spectral estimates based on spectral Ricci curvature assumptions

TL;DR

This paper establishes geometric and spectral estimates on Riemannian manifolds, deriving a Bonnet-Myers type theorem and isoperimetric inequalities under spectral Ricci curvature conditions.

Contribution

It introduces new spectral conditions involving the Ricci tensor that imply geometric properties like finiteness of the fundamental group and isoperimetric inequalities.

Findings

Finite fundamental group under spectral Ricci conditions

Isoperimetric inequalities derived from Kato conditions

Kato condition for Ricci curvature under geometric assumptions

Abstract

We obtain a Bonnet-Myers theorem under a spectral condition: a closed Riemannian manifold $(M^n,g)$ for which the lowest eigenvalue of the Ricci tensor $\rho$ is such that the Schr\"odinger operator $(n-2)\Delta + \rho$ is positive has finite fundamental group. As a continuation of our earlier results, we obtain isoperimetric inequalities from a Kato condition on the Ricci curvature. Furthermore, we obtain the Kato condition for the Ricci curvature under purely geometric assumptions.