Some rigidity theorems for spectral curvature bounds

TL;DR

This paper extends classical rigidity results to spectral curvature bounds, using warped μ-bubble methods to classify hypersurfaces and establish geometric estimates in 3-manifolds.

Contribution

It introduces spectral versions of classical curvature theorems, including classification, band width estimates, and splitting results under spectral curvature conditions.

Findings

Classification of stable weighted minimal hypersurfaces in 3-manifolds with nonnegative spectral scalar curvature

Band width estimates for spectral Ricci and spectral scalar curvatures

Spectral Geroch conjecture and Milnor conjecture related results

Abstract

We investigate the geometric implications of spectral curvature bounds, extending classical rigidity results in scalar curvature geometry to the spectral setting. By systematically employing the warped $\mu$-bubble method, we show classification theorems for stable weighted minimal hypersurfaces in 3-manifolds with nonnegative spectral scalar curvature, and we establish band width estimates for both spectral Ricci and spectral scalar curvatures. Furthermore, we prove some splitting theorems under spectral curvature conditions, including a spectral version of the Geroch conjecture for manifolds with arbitrary ends and a result related to the Milnor conjecture.