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Transactions of the American Mathematical Society

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

Author: Sławomir Kolasiński
Journal: Trans. Amer. Math. Soc. 367 (2015), 775-811
MSC (2010): Primary 49Q10; Secondary 28A75, 49Q20, 49Q15
Published electronically: July 24, 2014
MathSciNet review: 3280027
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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 \le l \le m+2$ and an $ m$-dimensional set $ \Sigma \subset R^n$, we also introduce global versions of this discrete curvature by taking the supremum with respect to $ (m+2-l)$ points on $ \Sigma $. 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 $ \Sigma $. Next, we prove that if $ \Sigma $ is compact and $ m$-Ahlfors regular and if $ p$ is greater than the dimension of the set of all $ l$-tuples of points on $ \Sigma $ (i.e. $ p > ml$), then the P. Jones' $ \beta $-numbers of $ \Sigma $ must decay as $ r^{\tau }$ with $ r \to 0$ for some $ \tau \in (0,1)$. If $ \Sigma $ is an immersed $ C^1$ manifold or a bilipschitz image of such a 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,\tau }$ manifold. We also define a wide class of other sets for which this assertion is true. After that, we bootstrap the exponent $ \tau $ to  $ \alpha = 1 - ml/p$, which is optimal due to our theorem with S. Blatt [Adv. Math., 2012]. This gives an analogue of the Morrey-Sobolev embedding theorem $ W^{2,p}(\mathbb{R}^{ml}) \subseteq C^{1,\alpha }(\mathbb{R}^{ml})$ but, more importantly, we also obtain a qualitative control over the local graph representations of $ \Sigma $ only in terms of the energy.

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Additional Information

Sławomir Kolasiński
Affiliation: Institute of Mathematics, University of Warsaw, Banacha 2, 02-097 Warsaw, Poland
Address at time of publication: Max Planck Institute for Gravitational Physics, Albert Einstein Institute, Am Mühlenberg 1, D-14476 Golm, Germany

Keywords: Menger curvature, Ahlfors regularity, repulsive potentials, regularity theory
Received by editor(s): May 24, 2012
Published electronically: July 24, 2014
Additional Notes: The major part of this work was accomplished while the author was working at the University of Warsaw and was supported by the Polish Ministry of Science grant no. N N201 611140. The work was put in its final form at the AEI Golm, AEI publication number AEI-2013-165
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.