Obata’s Rigidity Theorem for Metric Measure Spaces

Abstract

We prove Obata’s rigidity theorem for metric measure spaces that satisfy a Riemannian curvaturedimension condition. Additionally,we show that a lower bound K for the generalizedHessian 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.

Keywords

• eigenvalue
• suspension
• hessian
• convexity