Geometric Sobolev-like embedding using high-dimensional Menger-like curvature
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- by Sławomir Kolasiński PDF
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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.References
- Simon Blatt and Sławomir Kolasiński, Sharp boundedness and regularizing effects of the integral Menger curvature for submanifolds, Adv. Math. 230 (2012), no. 3, 839–852. MR 2921162, DOI 10.1016/j.aim.2012.03.007
- Simon Blatt, A note on integral Menger curvature for curves, Math. Nachr. 286 (2013), no. 2-3, 149–159. MR 3021472, DOI 10.1002/mana.201100220
- Guy David, Carlos Kenig, and Tatiana Toro, Asymptotically optimally doubling measures and Reifenberg flat sets with vanishing constant, Comm. Pure Appl. Math. 54 (2001), no. 4, 385–449. MR 1808649, DOI 10.1002/1097-0312(200104)54:4<385::AID-CPA1>3.0.CO;2-M
- Guy David and Stephen Semmes, Analysis of and on uniformly rectifiable sets, Mathematical Surveys and Monographs, vol. 38, American Mathematical Society, Providence, RI, 1993. MR 1251061, DOI 10.1090/surv/038
- Herbert Federer, Curvature measures, Trans. Amer. Math. Soc. 93 (1959), 418–491. MR 110078, DOI 10.1090/S0002-9947-1959-0110078-1
- O. Gonzalez, J. H. Maddocks, F. Schuricht, and H. von der Mosel, Global curvature and self-contact of nonlinearly elastic curves and rods, Calc. Var. Partial Differential Equations 14 (2002), no. 1, 29–68. MR 1883599, DOI 10.1007/s005260100089
- Oscar Gonzalez and John H. Maddocks, Global curvature, thickness, and the ideal shapes of knots, Proc. Natl. Acad. Sci. USA 96 (1999), no. 9, 4769–4773. MR 1692638, DOI 10.1073/pnas.96.9.4769
- Allen Hatcher, Algebraic topology, Cambridge University Press, Cambridge, 2002. MR 1867354
- Morris W. Hirsch, Differential topology, Graduate Texts in Mathematics, No. 33, Springer-Verlag, New York-Heidelberg, 1976. MR 0448362
- Peter W. Jones, The traveling salesman problem and harmonic analysis, Publ. Mat. 35 (1991), no. 1, 259–267. Conference on Mathematical Analysis (El Escorial, 1989). MR 1103619, DOI 10.5565/PUBLMAT_{3}5191_{1}2
- S. Kolasiński, P. Strzelecki, and H. von der Mosel, Compactness for the class of manifolds with equibounded curvature energy, in preparation.
- Sławomir Kolasiński, PawełStrzelecki, and Heiko von der Mosel, Characterizing $W^{2,p}$ submanifolds by $p$-integrability of global curvatures, Geom. Funct. Anal. 23 (2013), no. 3, 937–984. MR 3061777, DOI 10.1007/s00039-013-0222-y
- Sławomir Kolasiński, Integral Menger curvature for sets of arbitrary dimension and codimension. PhD thesis, Institute of Mathematics, University of Warsaw, 2011, arXiv:1011.2008.
- Sławomir Kolasiński and Marta Szumańska, Minimal Hölder regularity implying finiteness of integral Menger curvature, Manuscripta Math. 141 (2013), no. 1-2, 125–147. MR 3042684, DOI 10.1007/s00229-012-0565-y
- J. C. Léger, Menger curvature and rectifiability, Ann. of Math. (2) 149 (1999), no. 3, 831–869. MR 1709304, DOI 10.2307/121074
- Gilad Lerman and J. Tyler Whitehouse, High-dimensional Menger-type curvatures. II. $d$-separation and a menagerie of curvatures, Constr. Approx. 30 (2009), no. 3, 325–360. MR 2558685, DOI 10.1007/s00365-009-9073-z
- Gilad Lerman and J. Tyler Whitehouse, High-dimensional Menger-type curvatures. Part I: Geometric multipoles and multiscale inequalities, Rev. Mat. Iberoam. 27 (2011), no. 2, 493–555. MR 2848529, DOI 10.4171/RMI/645
- L. S. Pontryagin, Selected works. Vol. 3, Classics of Soviet Mathematics, Gordon & Breach Science Publishers, New York, 1986. Algebraic and differential topology; Edited and with a preface by R. V. Gamkrelidze; Translated from the Russian by P. S. V. Naidu. MR 898008
- E. R. Reifenberg, Solution of the Plateau Problem for $m$-dimensional surfaces of varying topological type, Acta Math. 104 (1960), 1–92. MR 114145, DOI 10.1007/BF02547186
- Sebastian Scholtes, For which positive $p$ is the integral Menger curvature $\mathcal {M}_{p}$ finite for all simple polygons?, 2012, arXiv:1202.0504.
- Sebastian Scholtes, Tangency properties of sets with finite geometric curvature energies, Fund. Math. 218 (2012), no. 2, 165–191. MR 2957689, DOI 10.4064/fm218-2-4
- Leon Simon, Reifenberg’s topological disc theorem, 1996. Mathematisches Institut Universität Tübingen. Preprints AB Analysis.
- PawełStrzelecki, Marta Szumańska, and Heiko von der Mosel, A geometric curvature double integral of Menger type for space curves, Ann. Acad. Sci. Fenn. Math. 34 (2009), no. 1, 195–214. MR 2489022
- Pawel Strzelecki, Marta Szumańska, and Heiko von der Mosel, Regularizing and self-avoidance effects of integral Menger curvature, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 9 (2010), no. 1, 145–187. MR 2668877
- PawełStrzelecki and Heiko von der Mosel, On rectifiable curves with $L^p$-bounds on global curvature: self-avoidance, regularity, and minimizing knots, Math. Z. 257 (2007), no. 1, 107–130. MR 2318572, DOI 10.1007/s00209-007-0117-4
- PawełStrzelecki and Heiko von der Mosel, Integral Menger curvature for surfaces, Adv. Math. 226 (2011), no. 3, 2233–2304. MR 2739778, DOI 10.1016/j.aim.2010.09.016
- PawełStrzelecki and Heiko von der Mosel, Tangent-point repulsive potentials for a class of non-smooth $m$-dimensional sets in $\Bbb {R}^n$. Part I: Smoothing and self-avoidance effects, J. Geom. Anal. 23 (2013), no. 3, 1085–1139. MR 3078345, DOI 10.1007/s12220-011-9275-z
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
- Email: s.kolasinski@mimuw.edu.pl
- 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
- © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 367 (2015), 775-811
- MSC (2010): Primary 49Q10; Secondary 28A75, 49Q20, 49Q15
- DOI: https://doi.org/10.1090/S0002-9947-2014-05989-8
- MathSciNet review: 3280027