Rigidity phenomena in manifolds with boundary under a lower weighted Ricci curvature bound

Abstract

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.

34 pages. arXiv admin note: text overlap with arXiv:1506.03223

References in corpus (2)

• On the geometry of Riemannian manifolds with density
• Rigidity of manifolds with boundary under a lower Bakry-E'mery Ricci curvature bound

Cited by in corpus (3)

• Concentration of $1$-Lipschitz functions on manifolds with boundary with Dirichlet boundary condition
• Rigidity of manifolds with boundary under a lower Bakry-E'mery Ricci curvature bound
• Comparison geometry of manifolds with boundary under a lower weighted Ricci curvature bound