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  Geometric Sobolev-like embedding using high-dimensional Menger-like curvature

Kolasinski, S. (2015). Geometric Sobolev-like embedding using high-dimensional Menger-like curvature. Transactions of the American Mathematical Society, 367(2), 775-811. doi:10.1090/S0002-9947-2014-05989-8.

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1205.4112 (Preprint), 488KB
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 Creators:
Kolasinski, Slawomir1, 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, Functional Analysis, math.FA,
 Abstract: We study a modified version of Lerman-Whitehouse Menger-like curvature defined for m+2 points in an n-dimensional Euclidean space. For 1 <= l <= m+2 and an m-dimensional subset S of R^n we also introduce global versions of this discrete curvature, by taking supremum with respect to m+2-l points on S. We then define geometric curvature energies by integrating one of the global Menger-like curvatures, raised to a certain power p, over all l-tuples of points on S. Next, we prove that if S is compact and m-Ahlfors regular and if p is greater than ml, then the P. Jones' \beta-numbers of S must decay as r^t with r \to 0 for some t in (0,1). If S is an immersed C^1 manifold or a bilipschitz image of such set then it follows that it is Reifenberg flat with vanishing constant, hence (by a theorem of David, Kenig and Toro) an embedded C^{1,t} manifold. We also define a wide class of other sets for which this assertion is true. After that, we bootstrap the exponent t to the optimal one a = 1 - ml/p showing an analogue of the Morrey-Sobolev embedding theorem. Moreover, we obtain a qualitative control over the local graph representations of S only in terms of the energy.

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 Dates: 2012-05-182013-03-0820142015
 Publication Status: Issued
 Pages: I removed Example 3.11, which was wrong in the sense that the \beta-numbers for this set do not decay as r^2
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 Rev. Type: -
 Identifiers: arXiv: 1205.4112
DOI: 10.1090/S0002-9947-2014-05989-8
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Title: Transactions of the American Mathematical Society
  Other : Trans. Amer. Math. Soc.
Source Genre: Journal
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Publ. Info: Providence, R.I. : American Mathematical Society
Pages: - Volume / Issue: 367 (2) Sequence Number: - Start / End Page: 775 - 811 Identifier: ISSN: 0002-9947
CoNE: https://pure.mpg.de/cone/journals/resource/954925218003