Inradius collapsed manifolds

TL;DR

This paper investigates the geometric and topological structure of manifolds with boundary undergoing inradius collapse under curvature and boundary bounds, characterizing their limit spaces and boundary component counts.

Contribution

It provides a comprehensive analysis of inradius collapsed manifolds with boundary, including their limit Alexandrov spaces and topological classification in various cases.

Findings

Limit spaces are Alexandrov spaces with curvature bounds.

Topology of inradius collapsed manifolds is characterized by singular I-bundles.

Number of boundary components is at most two, with conditions for disconnected boundaries.

Abstract

In this paper, we study collapsed manifolds with boundary, where we assume a lower sectional curvature bound, two sides bounds on the second fundamental forms of boundaries and upper diameter bound. Our main concern is the case when inradii of manifolds converge to zero. This is a typical case of collapsing manifolds with boundary. We determine the limit spaces of inradius collapsed manifolds as Alexandrov spaces with curvature uniformly bounded below. When the limit space has co-dimension one, we completely determined the topology of inradius collapsed manifold in terms of singular $I$-bundles. Genral inradius collapse to almost regular spaces are also characterized. In the general case of unbounded diameters, we prove that the number of boundary components of inradius collapsed manifolds is at most two, where the disconnected boundary happens if and only if the manifold has a topological product structure.