SEQUENCES OF OPEN RIEMANNIAN MANIFOLDS WITH BOUNDARY

摘要

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 δ-inner regions, that avoid the boundary by a distance δ. We prove Gromov-Hausdorff compactness theorems for sequences of these δ-inner regions. We then build "glued limit spaces" out of the Gromov-Hausdorff limits of δ-inner 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.