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Characterizing W2 p submanifolds by p-integrability of global curvatures

Kolasinski, S., Strzelecki, P., & von der Mosel, H. (2013). Characterizing W2 p submanifolds by p-integrability of global curvatures. Geometric and Functional Analysis, 23(3), 937-984. doi:10.1007/s00039-013-0222-y.

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1203.4688 (Preprint), 807KB
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Creators:
Kolasinski, Slawomir1, Author
Strzelecki, Paweł, Author
von der Mosel, Heiko, Author
Affiliations:
1Geometric Measure Theory, AEI-Golm, MPI for Gravitational Physics, Max Planck Society, ou_1753352

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Free keywords: Mathematics, Classical Analysis and ODEs, math.CA,Mathematics, Functional Analysis, math.FA,
Abstract: We give sufficient and necessary geometric conditions, guaranteeing that an immersed compact closed manifold $\Sigma^m\subset \R^n$ of class $C^1$ and of arbitrary dimension and codimension (or, more generally, an Ahlfors-regular compact set $\Sigma$ satisfying a mild general condition relating the size of holes in $\Sigma$ to the flatness of $\Sigma$ measured in terms of beta numbers) is in fact an embedded manifold of class $C^{1,\tau}\cap W^{2,p}$, where $p>m$ and $\tau=1-m/p$. The results are based on a careful analysis of Morrey estimates for integral curvature--like energies, with integrands expressed geometrically, in terms of functions that are designed to measure either (a) the shape of simplices with vertices on $\Sigma$ or (b) the size of spheres tangent to $\Sigma$ at one point and passing through another point of $\Sigma$. Appropriately defined \emph{maximal functions} of such integrands turn out to be of class $L^p(\Sigma)$ for $p>m$ if and only if the local graph representations of $\Sigma$ have second order derivatives in $L^p$ and $\Sigma$ is embedded. There are two ingredients behind this result. One of them is an equivalent definition of Sobolev spaces, widely used nowadays in analysis on metric spaces. The second one is a careful analysis of local Reifenberg flatness (and of the decay of functions measuring that flatness) for sets with finite curvature energies. In addition, for the geometric curvature energy involving tangent spheres we provide a nontrivial lower bound that is attained if and only if the admissible set $\Sigma$ is a round sphere.

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Dates: 2012-03-212012-10-202013
Publication Status: Published in print
Pages: 44 pages, 2 figures; several minor corrections
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Identifiers: arXiv: 1203.4688
DOI: 10.1007/s00039-013-0222-y
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Title: Geometric and Functional Analysis
Source Genre: Journal
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Pages: - Volume / Issue: 23 (3) Sequence Number: - Start / End Page: 937 - 984 Identifier: -