The Journal of Geometric Analysis

, Volume 29, Issue 1, pp 1–32 | Cite as

Rigidity Phenomena in Manifolds with Boundary Under a Lower Weighted Ricci Curvature Bound

  • Yohei SakuraiEmail author


We study Riemannian manifolds with boundary under a lower N-weighted Ricci curvature bound for N at most 1, and under a lower weighted mean curvature bound for the boundary. We examine rigidity phenomena in such manifolds with boundary. We conclude a volume growth rigidity theorem for the metric neighborhoods of the boundaries, and various splitting theorems. We also obtain rigidity theorems for the smallest Dirichlet eigenvalues for the weighted p-Laplacians.


Manifold with boundary Weighted Ricci curvature Weighted p-Laplacian 

Mathematics Subject Classification

Primary 53C20 



The author would like to express his gratitude to Professor Koichi Nagano for his constant advice and suggestions. The author would also like to thank Professor Shin-ichi Ohta for his valuable comments. The author would like to thank Professor William Wylie for his valuable advice concerning Proposition 2.9.


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Authors and Affiliations

  1. 1.Advanced Institute for Materials ResearchTohoku UniversitySendaiJapan

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