Collapsing Manifolds with Boundary

TL;DR

This paper investigates how manifolds with boundary collapse in the Gromov-Hausdorff topology, revealing structural and limit space properties under curvature and boundary bounds.

Contribution

It establishes a disc bundle structure for manifolds with boundary under curvature and boundary bounds, and characterizes their Gromov-Hausdorff limits as Alexandrov spaces.

Findings

Manifolds with boundary close to a closed manifold have a disc bundle structure.

Limits of certain boundary manifolds are Alexandrov spaces with curvature bounds.

Provides conditions under which collapsing manifolds have well-understood limit spaces.

Abstract

This manuscript studies manifolds-with-boundary collapsing in the Gromov-Hausdorff topology. The main aim is an understanding of the relationship of the topology and geometry of a limiting sequence of manifolds-with-boundary to that of a limit space, which is presumed to be without geodesic terminals. The main result establishes a disc bundle structure for any manifold-with-boundary having two-sided bounds on sectional curvature and second fundamental form, and a lower bound on intrinsic injectivity radius, which is sufficiently close in the Gromov-Hausdorff topology to a closed manifold. The second main result identifies Gromov-Hausdorff limits of certain sequences of manifolds-with-boundary as Alexandrov spaces of curvature bounded below.