Topology of Riemannian submanifolds with prescribed boundary
type: Item
instance of: scholarly article
title: Topology of Riemannian submanifolds with prescribed boundary
zbMATH Open document ID: 1201.53005
DOI: 10.1215/00127094-2010-018
author: Stephanie B. Alexander
author: Mohammad Ghomi
author: Jeremy Wong
published in: Duke Mathematical Journal
publication date: 2010-05-17T00:00:00Z
review text: In this interesting paper, the authors prove that a smooth compact submanifold of codimension \(2\) immersed in \(\mathbb R ^{n}, \, n \geq 3,\) bounds at most finitely many topologically distinct, compact, nonnegatively curved hypersurfaces. This solves a question of \textit{B. Guan} and \textit{J. Spruck} [Commun. Pure Appl. Math. 57, No. 10, 1311--1331 (2004; Zbl 1066.53109)] related to a problem of Yau. Analogous results for complete fillings of arbitrary Riemannian submanifolds are also obtained. They also show that these finiteness theorems may not hold if the codimension is too high or the prescribed boundary is not sufficiently regular. The methods of the proofs include a relative version of Nash's isometric embedding theorem and the theory of Alexandrov spaces with curvature bounded below, including the compactness and stability theorems of Gromov and Perelman.
reviewed by: Maria Aparecida Soares Ruas
Mathematics Subject Classification ID: 53A07
Mathematics Subject Classification ID: 53C23
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 53C45
zbMATH DE Number: 5708812
zbMATH Keywords: Riemannian submanifolds
zbMATH Keywords: negative curvature
zbMATH Keywords: locally convex hypersurfaces
zbMATH Keywords: topological types.
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1215/00127094-2010-018
OpenAlex ID: W1994862240
Wikidata QID: Q115240687
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description: scientific article; zbMATH DE number 5708812