Compactness of immersions with local Lipschitz representation
type: Item
instance of: scholarly article
title: Compactness of immersions with local Lipschitz representation
zbMATH Open document ID: 1254.53010
author: Patrick Breuning
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 2012-08-24T00:00:00Z
full work available at URL: https://arxiv.org/abs/1201.4993
reviewed by: Juan Monterde
Mathematics Subject Classification ID: 53A07
Mathematics Subject Classification ID: 53A05
Mathematics Subject Classification ID: 58D10
zbMATH DE Number: 6071658
zbMATH Keywords: Lipschitz graph
zbMATH Keywords: compactness of sets of immersed surfaces
MaRDI profile type: Publication
OpenAlex ID: W3101080243
arXiv ID: 1201.4993
cites work: Q3148988
cites work: Grassmann manifolds and the Grassmann image of submanifolds
cites work: Q3017788
cites work: Q3946815
cites work: Immersions with bounded curvature
cites work: Q5538378
cites work: How to conjugate C\(^1\)-close group actions
cites work: An exact sequence in differential topology
cites work: Riemannian geometry and geometric analysis
cites work: Bernstein type theorems for higher codimension
cites work: Riemannian center of mass and mollifier smoothing
cites work: A compactness theorem for surfaces with \(L_ p\)-bounded second fundamental form
cites work: Zur Riemannschen Geometrie in Grassmannschen Mannigfaltigkeiten
cites work: SECTIONAL CURVATURES OF GRASSMANN MANIFOLDS
DOI: 10.1016/J.ANIHPC.2012.02.001
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review text: The results in the paper under review are in the spirit of Cheeger's finiteness theorem. They are a natural generalization which can be traced back to \textit{J. Langer} [Math. Ann. 270, 223--234 (1985; Zbl 0564.58010)] where the compactness of immersed surfaces in \({\mathbb R}^3\) admitting uniform bounds on the second fundamental form and the area of the surfaces is studied. The finiteness of topological types was generalized to arbitrary dimension and codimension by \textit{K. Corlette} [Geom. Dedicata 33, No. 2, 153--161 (1990; Zbl 0717.53035)] and the compactness theorem was generalized to hypersurfaces of arbitrary dimension by \textit{S. Delladio} [J. Geom. Anal. 11, No. 1, 17--42 (2001; Zbl 1034.49043)].NEWLINENEWLINELet \(\mathcal{F}^1_v(r,\lambda)\) be the set of \(C^1\)-immersions \(f:M^m\to {\mathbb R}^{m+1}\) which may locally be written over an \(m\)-space as the graph of a \(\lambda\)-Lipschitz function \(u:B_r\to{\mathbb R}\), and with vol\((M)\leq v\). Analogously, \(\mathcal{F}^0(r,\lambda)\) denotes the set of Lipschitz immersions.NEWLINENEWLINEThe first main result is the compactness of \((r,\lambda)\)-immersions in codimension one. \(\mathcal{F}^1_v(r,\lambda)\) is relatively compact in \(\mathcal{F}^0(r,\lambda)\) in the following sense: For a any sequence \(f^i:M^i\to {\mathbb R}^{m+1}\) there exist a subsequence \(f^{i_k}\), an \(f:M\to {\mathbb R}^{m+1}\) in \(\mathcal{F}^0(r,\lambda)\), and diffeomorphisms \(\phi^{i_k}:M\to M^{i_k}\), such that \(f^{i_k}\circ\phi_{i_k}\) is uniformly Lipschitz bounded and converges uniformly to \(f\). Moreover, up to diffeomorphism, there are only finitely many manifolds in \(\mathcal{F}^1_v(r,\lambda)\).NEWLINENEWLINENext, the compactness of \((r,\lambda)\)-immersions in arbitrary codimension is shown, but only for \(\lambda\leq\frac14\). In the end of the last section the author discuss some possibilities of how to prove the same result for a bigger Lipschitz constant.
description: scientific article; zbMATH DE number 6071658