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A Characterization of Codimension One Collapse Under Bounded Curvature and Diameter

Let [Formula: see text] be the space of closed n-dimensional Riemannian manifolds (M, g) with [Formula: see text] and [Formula: see text] . In this paper we consider sequences [Formula: see text] in [Formula: see text] converging in the Gromov–Hausdorff topology to a compact metric space Y. We show,...

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Autor principal: Roos, Saskia
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Springer US 2017
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6294179/
https://www.ncbi.nlm.nih.gov/pubmed/30839898
http://dx.doi.org/10.1007/s12220-017-9930-0
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author Roos, Saskia
author_facet Roos, Saskia
author_sort Roos, Saskia
collection PubMed
description Let [Formula: see text] be the space of closed n-dimensional Riemannian manifolds (M, g) with [Formula: see text] and [Formula: see text] . In this paper we consider sequences [Formula: see text] in [Formula: see text] converging in the Gromov–Hausdorff topology to a compact metric space Y. We show, on the one hand, that the limit space of this sequence has at most codimension one if there is a positive number r such that the quotient [Formula: see text] can be uniformly bounded from below by a positive constant C(n, r, Y) for all points [Formula: see text] . On the other hand, we show that if the limit space has at most codimension one then for all positive r there is a positive constant C(n, r, Y) bounding the quotient [Formula: see text] uniformly from below for all [Formula: see text] . As a conclusion, we derive a uniform lower bound on the volume and a bound on the essential supremum of the sectional curvature for the closure of the space consisting of all manifolds in [Formula: see text] with [Formula: see text] .
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spelling pubmed-62941792018-12-28 A Characterization of Codimension One Collapse Under Bounded Curvature and Diameter Roos, Saskia J Geom Anal Article Let [Formula: see text] be the space of closed n-dimensional Riemannian manifolds (M, g) with [Formula: see text] and [Formula: see text] . In this paper we consider sequences [Formula: see text] in [Formula: see text] converging in the Gromov–Hausdorff topology to a compact metric space Y. We show, on the one hand, that the limit space of this sequence has at most codimension one if there is a positive number r such that the quotient [Formula: see text] can be uniformly bounded from below by a positive constant C(n, r, Y) for all points [Formula: see text] . On the other hand, we show that if the limit space has at most codimension one then for all positive r there is a positive constant C(n, r, Y) bounding the quotient [Formula: see text] uniformly from below for all [Formula: see text] . As a conclusion, we derive a uniform lower bound on the volume and a bound on the essential supremum of the sectional curvature for the closure of the space consisting of all manifolds in [Formula: see text] with [Formula: see text] . Springer US 2017-10-03 2018 /pmc/articles/PMC6294179/ /pubmed/30839898 http://dx.doi.org/10.1007/s12220-017-9930-0 Text en © The Author(s) 2017 Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
spellingShingle Article
Roos, Saskia
A Characterization of Codimension One Collapse Under Bounded Curvature and Diameter
title A Characterization of Codimension One Collapse Under Bounded Curvature and Diameter
title_full A Characterization of Codimension One Collapse Under Bounded Curvature and Diameter
title_fullStr A Characterization of Codimension One Collapse Under Bounded Curvature and Diameter
title_full_unstemmed A Characterization of Codimension One Collapse Under Bounded Curvature and Diameter
title_short A Characterization of Codimension One Collapse Under Bounded Curvature and Diameter
title_sort characterization of codimension one collapse under bounded curvature and diameter
topic Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6294179/
https://www.ncbi.nlm.nih.gov/pubmed/30839898
http://dx.doi.org/10.1007/s12220-017-9930-0
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