Volume 23, issue 6 (2019)

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Inradius collapsed manifolds

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.

Keywords
collapse, Gromov–Hausdorff convergence, manifold with boundary, inradius
Mathematical Subject Classification 2010
Primary: 53C20, 53C21, 53C23
References
Publication
Received: 9 December 2015
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