A Compactness Theorem for Riemannian Manifolds with Boundary and Applications

2012/11/27 by Knox, Kenneth S.
#53C20 #53C21 #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1211.6210

Abstract

In this paper we prove weak L^1,p (and thus C^α) compactness for the class of uniformly mean-convex Riemannian n-manifolds with boundary satisfying bounds on curvature quantities, diameter, and (n-1)-volume of the boundary. We obtain two stability theorems from the compactness result. The first theorem applies to 3-manifolds (contained in the aforementioned class) that have Ricci curvature close to 0 and whose boundaries are Gromov-Hausdorff close to a fixed metric on S² with positive curvature. Such manifolds are C^α close to the region enclosed by a Weyl embedding of the fixed metric into \R³. The second theorem shows that a 3-manifold with Ricci curvature close to 0 (resp. -2, 2) and mean curvature close to 2 (resp. 2√ 2, 0) is C^α close to a metric ball in the space form of constant curvature 0 (resp -1, 1), provided that the boundary is a topological sphere.

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