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.

Comparison Geometry of Manifolds with Boundary under a Lower Weighted Ricci Curvature Bound

“…for every z ∈ ∂M . Under the conditions (1.2) and (1.3) for κ, λ ∈ R, N ∈ (−∞, 1], we formulate various comparison geometric results, and generalize the preceding studies by Kasue [9], [10], and the author [17] when f = 0.…”

“…When κ = 0 and λ = 0, Theorem 1.1 was proven by the author in the cases N ∈ [n, ∞] (see [18]) and N ∈ (−∞, 1] (see [19]). In the unweighted case of f = 0, Kasue [9] has proved Theorem 1.1 under the assumption that M is non-compact and ∂M is compact (see also Croke and Kleiner [5]), and the author [17] has proved Theorem 1.1 itself.…”

Well‐posedness of the Laplacian on manifolds with boundary and bounded geometry

“…for all ∈ 1, loc ( ). In a nice, very recent paper [62], Sakurai has proved this Poincaré inequality for = ∅, and = 1. His proof (posted on the Arxive preprint server shortly before we posted the first version this paper) can be extended to our case (once one takes care of a few delicate points; see Remarks 3.2 and 3.3).…”

“…Nevertheless, this problem can be circumvented by first assuming that the metric has a product structure near the boundary. The necessary preliminaries for this part will be given in Section 3.3 and the proof of our Poincaré inequality in this case is carried out in Subsection 3.4, following the method of [62]. The case of a general metric is then obtained by comparing the given metric with a metric that has a product structure near the boundary.…”

“…Proposition 3.7 (Special case of [62,Lemma 4.5]). We assume that is a manifold with boundary and Ricci curvature bounded from below.…”