Obata's rigidity theorem for metric measure spaces

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.

Theorem 6.1 (Corollary 6.2) is not needed anymore for the proof. Corollary 6.2 has become Corollar 5.4. A rigidity result concerning higher eigenvalues has been added. 20p. Comments are welcome

References in corpus (2)

• Self-improvement of the Bakry-Émery condition and Wasserstein contraction of the heat flow in RCD(K,\infty) metric measure spaces
• Sharp geometric and functional inequalities in metric measure spaces with lower Ricci curvature bounds

Cited by in corpus (14)

• Sharp geometric and functional inequalities in metric measure spaces with lower Ricci curvature bounds
• On the topology and the boundary of N-dimensional RCD(K,N) spaces
• Quantitative Obata's Theorem
• The behavior of harmonic functions at singular points of $\mathsf{RCD}$ spaces
• Maximal volume entropy rigidity for ${\mathsf{RCD}}^{\ast }(-(N-1),N)$ spaces
• The equality case in Cheeger's and Buser's inequalities on $\mathsf{RCD}$ spaces
• Rigidity for the spectral gap on $RCD(K,\infty )$-spaces
• Inscribed Radius Bounds for Lower Ricci Bounded Metric Measure Spaces with Mean Convex Boundary
• Rigidity of manifolds with boundary under a lower Bakry-E'mery Ricci curvature bound
• Sphere theorems for RCD and stratified spaces
• Spectral convergence of high-dimensional spheres to Gaussian spaces
• Characterization of equality in Zhong-Yang type (sharp) spectral gap estimates for metric measure spaces
• Lower Bound Estimates for The First Eigenvalue of The Weighted $p$-Laplacian on Smooth Metric Measure Spaces
• Convergence to the Product of the Standard Spheres and Eigenvalues of the Laplacian