Sequences of Open Riemannian Manifolds with Boundary

TL;DR

This paper develops a framework for analyzing limits of sequences of open Riemannian manifolds with boundary by using inner regions and glued limit spaces, with applications to noncollapsing sequences with nonnegative Ricci curvature.

Contribution

It introduces a new approach to define and study limits of open Riemannian manifolds with boundary using $ ext{delta}$ inner regions and glued limit spaces, extending Gromov-Hausdorff compactness theory.

Findings

Gromov-Hausdorff compactness theorems for $ ext{delta}$ inner regions.

Construction of glued limit spaces from these regions.

Application to sequences with nonnegative Ricci curvature.

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.