Cheng's Theorem in the Spectral Setting

Abstract

We prove a maximal diameter rigidity theorem for closed 3-dimensional Riemannian manifolds under a spectral lower bound for the Ricci curvature. We show that if $0<\gamma <2$, $K>0$ and ${\lambda }_1(-\gamma \Delta +\mathrm{Ric})\ge 2K$, then the optimal bound $\mathrm{diam}(M,g)\le \frac{\pi }{\sqrt{K}}$ is attained if and only if (M,g) is isometric to the round 3-sphere of constant sectional curvature $K$.