Rigidity of manifolds with boundary under a lower Ricci curvature bound

Abstract

We study Riemannian manifolds with boundary under a lower Ricci curvature bound, and a lower mean curvature bound for the boundary. We prove a volume comparison theorem of Bishop-Gromov type concerning the volumes of the metric neighborhoods of the boundaries. We conclude several rigidity theorems. As one of them, we obtain a volume growth rigidity theorem. We also show a splitting theorem of Cheeger-Gromoll type under the assumption of the existence of a single ray.

43 pages

References in corpus (1)

• Sharp diameter estimates for compact manifold with boundary

Cited by in corpus (2)

• Rigidity phenomena in manifolds with boundary under a lower weighted Ricci curvature bound
• Rigidity of manifolds with boundary under a lower Bakry-E'mery Ricci curvature bound