#### Volume 23, issue 6 (2019)

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### Takao Yamaguchi and Zhilang Zhang

Geometry & Topology 23 (2019) 2793–2860
##### Abstract

We study collapsed manifolds with boundary, where we assume a lower sectional curvature bound, two side 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 codimension one, we completely determine the topology of inradius collapsed manifold in terms of singular $I$–bundles. General 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.

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##### Keywords
collapse, Gromov–Hausdorff convergence, manifold with boundary, inradius
##### Mathematical Subject Classification 2010
Primary: 53C20, 53C21, 53C23
##### Publication
Revised: 31 January 2019
Accepted: 19 March 2019
Published: 1 December 2019
Proposed: Dmitri Burago
Seconded: Yasha Eliashberg, Tobias H Colding
##### Authors
 Takao Yamaguchi Department of Mathematics Kyoto University Kyoto Japan Zhilang Zhang School of Mathematics and Big Data Foshan University Foshan China Graduate School of Pure and Applied Sciences University of Tsukuba Tsukuba Japan