Sequences of Open Riemannian Manifolds with Boundary

Abstract

We consider sequences of open Riemannian manifolds with boundary that have no regularity conditions on the boundary. To define a reasonable notion of a limit of such a sequence, we examine "$\delta$ inner regions" which avoid the boundary by a distance $\delta$. We prove Gromov-Hausdorff compactness theorems for sequences of these "$\delta$ inner regions". We then build "glued limit spaces" out of the Gromov-Hausdorff limits of these $\delta$ interior regions and study the properties of these glued limit spaces. Our main applications assume the sequence is noncollapsing and has nonnegative Ricci curvature. We include open questions.

38 pages, 5 figures, [v2 changes: added citations to additional related work, new Defn 4.7 and Ex 4.13, new notion of "completed glued limit" added to Thm 6.3, Section 6.3 redone completely, Introduction changed accordingly, 40 pages]

Cited by in corpus (5)

• Convergence of Manifolds and Metric Spaces with Boundary
• Inradius collapsed manifolds
• Boundary harmonic coordinates on manifolds with boundary in low regularity
• Gromov-Hausdorff distance with boundary and its application to Gromov hyperbolic spaces and uniform spaces
• Precompactness of domains with lower Ricci curvature bound under Gromov-Hausdorff topology