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

2015/06/10 by Yohei Sakurai, Sakurai, Yohei · 3 citations
Mathematics · #Boundary (topology) #Curvature #Curvature of Riemannian manifolds #Differential Geometry (math.DG) #Dirichlet boundary condition #Dirichlet eigenvalue #Dirichlet's principle #Eigenvalues and eigenvectors #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry #Geometry and complex manifolds #Laplace operator #Mathematical analysis #Mathematics #Physics #Point processes and geometric inequalities #Pure mathematics #Ricci curvature #Rigidity (electromagnetism) #Scalar curvature #Sectional curvature #Upper and lower bounds #math.DG

paper · pdf · doi:10.48550/arxiv.1506.03223

published in arXiv (Cornell University) (Cornell University) · 48 pages. arXiv admin note: text overlap with arXiv:1404.3845

openalex publication_date 2015/06/10 · openalex created_date 2016/06/24 · arxiv created 2016/09/21 · arxiv updated 2016/09/22 · openalex updated_date 2026/08/05

Abstract

We study Riemannian manifolds with boundary under a lower Bakry-E'mery Ricci curvature bound. In our weighted setting, we prove several rigidity theorems for such manifolds with boundary. We conclude a rigidity theorem for the inscribed radii, a volume growth rigidity theorem for the metric neighborhoods of the boundaries, and various splitting theorems. We also obtain rigidity results for the smallest Dirichlet eigenvalues for the weighted p-Laplacians.

Citations

• Ricci curvature for metric-measure spaces via optimal transport
• A Picone's identity for the p-Laplacian and applications
• Normal and Integral Currents
• Lower Bounds on Ricci Curvature and the Almost Rigidity of Warped Products
• The splitting theorem in non-smooth context
• Rigidity of manifolds with boundary under a lower Ricci curvature bound

Cited by

• Concentration of 1-Lipschitz functions on manifolds with boundary with Dirichlet boundary condition
• Rigidity phenomena in manifolds with boundary under a lower weighted Ricci curvature bound

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