Sequences of open Riemannian manifolds with boundary

Raquel Perales and Christina Sormani

Vol. 270 (2014), No. 2, 423–471

DOI: 10.2140/pjm.2014.270.423

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, that 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 $\delta$-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.

Keywords

Gromov–Hausdorff, manifold, boundary

Mathematical Subject Classification 2010

Primary: 53C23

Milestones

Received: 8 February 2013

Revised: 18 June 2013

Accepted: 5 August 2013

Published: 22 August 2014