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  Compactness and isotopy finiteness for submanifolds with uniformly bounded geometric curvature energies

Kolasinski, S., Strzelecki, P., & von der Mosel, H. (2018). Compactness and isotopy finiteness for submanifolds with uniformly bounded geometric curvature energies. Communications in analysis and geometry, 26(6), 1251-1316. doi:10.4310/CAG.2018.v26.n6.a2.

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Item Permalink: http://hdl.handle.net/11858/00-001M-0000-0026-B8D2-2 Version Permalink: http://hdl.handle.net/21.11116/0000-0003-64DD-4
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 Creators:
Kolasinski, Slawomir1, Author              
Strzelecki, Paweł, Author
von der Mosel, Heiko, Author
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1Geometric Measure Theory, AEI-Golm, MPI for Gravitational Physics, Max Planck Society, ou_1753352              

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Free keywords: Mathematics, Differential Geometry, math.DG,Mathematics, Analysis of PDEs, math.AP,Mathematics, Metric Geometry, math.MG,
 Abstract: We prove isotopy finiteness for various geometric curvature energies including integral Menger curvature, and tangent-point repulsive potentials, defined on the class of compact, embedded $m$-dimensional Lipschitz submanifolds in ${\mathbb{R}}^n$. That is, there are only finitely many isotopy types of such submanifolds below a given energy value, and we provide explicit bounds on the number of isotopy types in terms of the respective energy. Moreover, we establish $C^1$-compactness: any sequence of submanifolds with uniformly bounded energy contains a subsequence converging in $C^1$ to a limit submanifold with the same energy bound. In addition, we show that all geometric curvature energies under consideration are lower semicontinuous with respect to Hausdorff-convergence, which can be used to minimise each of these energies within prescribed isotopy classes.

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 Dates: 2015-04-1720152018
 Publication Status: Published in print
 Pages: 44 pages, 5 figures
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 Rev. Method: -
 Identifiers: arXiv: 1504.04538
URI: http://arxiv.org/abs/1504.04538
DOI: 10.4310/CAG.2018.v26.n6.a2
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Title: Communications in analysis and geometry
Source Genre: Journal
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Pages: - Volume / Issue: 26 (6) Sequence Number: - Start / End Page: 1251 - 1316 Identifier: -