Obata's rigidity theorem for metric measure spaces

TL;DR

This paper extends Obata's rigidity theorem to metric measure spaces with curvature-dimension conditions, linking Hessian bounds to convexity and deriving eigenvalue rigidity results.

Contribution

It generalizes Obata's theorem to non-smooth spaces and establishes a characterization of Hessian bounds via convexity.

Findings

Rigidity result for metric measure spaces satisfying curvature-dimension conditions

Equivalence between lower Hessian bounds and K-convexity of functions

Rigidity for higher order eigenvalues in this setting

Abstract

We prove Obata's rigidity theorem for metric measure spaces that satisfy a Riemannian curvature-dimension condition. Additionally, we show that a lower bound $K$ for the generalized Hessian of a sufficiently regular function $u$ holds if and only if $u$ is $K$-convex. A corollary is also a rigidity result for higher order eigenvalues.