Sphere theorems and eigenvalue pinching without positive Ricci curvature assumption

TL;DR

This paper extends sphere theorems and eigenvalue pinching results to manifolds with Ricci curvature bounded below by a negative constant, removing the need for positive Ricci curvature assumptions, and only requiring bounds on Ricci and diameter.

Contribution

It generalizes existing sphere theorems by removing the positive Ricci curvature assumption, relying instead on Ricci lower bounds and diameter constraints.

Findings

Eigenvalue pinching results hold under Ricci lower bounds and diameter constraints.

Almost rigidity of the Obata theorem is established without positive Ricci curvature.

Generalization of Petersen and Aubry's sphere theorem to broader curvature conditions.

Abstract

Considering the almost rigidity of the Obata theorem, we generalize Petersen and Aubry's sphere theorem about eigenvalue pinching without assuming the positivity of Ricci curvature, only assuming $Ric\geq-Kg$ and $diam\leq D$ for some positive constants $K>0$ and $D>0$.