Integral Menger curvature and rectifiability of $n$-dimensional Borel sets in Euclidean $N$-space
HTML articles powered by AMS MathViewer
- by Martin Meurer PDF
- Trans. Amer. Math. Soc. 370 (2018), 1185-1250 Request permission
Abstract:
In this paper we show that an $n$-dimensional Borel set in Euclidean $N$-space with finite integral Menger curvature is $n$-rectifiable, meaning that it can be covered by countably many images of Lipschitz continuous functions up to a null set in the sense of Hausdorff measure. This generalises Léger’s rectifiability result for one-dimensional sets to arbitrary dimension and co-dimension. In addition, we characterise possible integrands and discuss examples known from the literature.
Intermediate results of independent interest include upper bounds of different versions of P. Jones’s $\beta$-numbers in terms of integral Menger curvature without assuming lower Ahlfors regularity, in contrast to the results of Lerman and Whitehouse [Constr. Approx. 30 (2009), 325–360].
References
- Robert A. Adams and John J. F. Fournier, Sobolev spaces, 2nd ed., Pure and Applied Mathematics (Amsterdam), vol. 140, Elsevier/Academic Press, Amsterdam, 2003. MR 2424078
- Jonas Azzam and Xavier Tolsa, Characterization of $n$-rectifiability in terms of Jones’ square function: Part II, Geom. Funct. Anal. 25 (2015), no. 5, 1371–1412. MR 3426057, DOI 10.1007/s00039-015-0334-7
- 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
- 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
- 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
- James J. Dudziak, Vitushkin’s conjecture for removable sets, Universitext, Springer, New York, 2010. MR 2676222, DOI 10.1007/978-1-4419-6709-1
- Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1158660
- K. J. Falconer, The geometry of fractal sets, Cambridge Tracts in Mathematics, vol. 85, Cambridge University Press, Cambridge, 1986. MR 867284
- Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
- Michael H. Freedman, Zheng-Xu He, and Zhenghan Wang, Möbius energy of knots and unknots, Ann. of Math. (2) 139 (1994), no. 1, 1–50. MR 1259363, DOI 10.2307/2946626
- Loukas Grafakos, Classical Fourier analysis, 2nd ed., Graduate Texts in Mathematics, vol. 249, Springer, New York, 2008. MR 2445437
- Immo Hahlomaa, Menger curvature and Lipschitz parametrizations in metric spaces, Fund. Math. 185 (2005), no. 2, 143–169. MR 2163108, DOI 10.4064/fm185-2-3
- Immo Hahlomaa, Curvature integral and Lipschitz parametrization in 1-regular metric spaces, Ann. Acad. Sci. Fenn. Math. 32 (2007), no. 1, 99–123. MR 2297880
- Immo Hahlomaa, Menger curvature and rectifiability in metric spaces, Adv. Math. 219 (2008), no. 6, 1894–1915. MR 2456269, DOI 10.1016/j.aim.2008.07.013
- Peter W. Jones, Rectifiable sets and the traveling salesman problem, Invent. Math. 102 (1990), no. 1, 1–15. MR 1069238, DOI 10.1007/BF01233418
- 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
- Sławomir Kolasiński, Geometric Sobolev-like embedding using high-dimensional Menger-like curvature, Trans. Amer. Math. Soc. 367 (2015), no. 2, 775–811. MR 3280027, DOI 10.1090/S0002-9947-2014-05989-8
- 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
- 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
- Yong Lin and Pertti Mattila, Menger curvature and $C^1$ regularity of fractals, Proc. Amer. Math. Soc. 129 (2001), no. 6, 1755–1762. MR 1814107, DOI 10.1090/S0002-9939-00-05814-7
- Pertti Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. Fractals and rectifiability. MR 1333890, DOI 10.1017/CBO9780511623813
- Karl Menger, Untersuchungen über allgemeine Metrik, Math. Ann. 103 (1930), no. 1, 466–501 (German). MR 1512632, DOI 10.1007/BF01455705
- Fedor Nazarov, Xavier Tolsa, and Alexander Volberg, The Riesz transform, rectifiability, and removability for Lipschitz harmonic functions, Publ. Mat. 58 (2014), no. 2, 517–532. MR 3264510
- Jun O’Hara, Energy of a knot, Topology 30 (1991), no. 2, 241–247. MR 1098918, DOI 10.1016/0040-9383(91)90010-2
- Sebastian Scholtes, For which positive $p$ is the integral Menger curvature $\mathcal {M}_{p}$ finite for all simple polygons?, 2012, arXiv:1202.0504.
- P. Stein, Classroom Notes: A Note on the Volume of a Simplex, Amer. Math. Monthly 73 (1966), no. 3, 299–301. MR 1533698, DOI 10.2307/2315353
- 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, Marta Szumańska, and Heiko von der Mosel, On some knot energies involving Menger curvature, Topology Appl. 160 (2013), no. 13, 1507–1529. MR 3091327, DOI 10.1016/j.topol.2013.05.022
- 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, Menger curvature as a knot energy, Phys. Rep. 530 (2013), no. 3, 257–290. MR 3105400, DOI 10.1016/j.physrep.2013.05.003
- 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
- Xavier Tolsa, Characterization of $n$-rectifiability in terms of Jones’ square function: part I, Calc. Var. Partial Differential Equations 54 (2015), no. 4, 3643–3665. MR 3426090, DOI 10.1007/s00526-015-0917-z
- Xavier Tolsa, Analytic capacity, the Cauchy transform, and non-homogeneous Calderón-Zygmund theory, Progress in Mathematics, vol. 307, Birkhäuser/Springer, Cham, 2014. MR 3154530, DOI 10.1007/978-3-319-00596-6
- Xavier Tolsa, Rectifiable measures, square functions involving densities, and the Cauchy transform, Mem. Amer. Math. Soc. 245 (2017), no. 1158, v+130. MR 3589161, DOI 10.1090/memo/1158
- Xavier Tolsa and Tatiana Toro, Rectifiability via a square function and Preiss’ theorem, Int. Math. Res. Not. IMRN 13 (2015), 4638–4662. MR 3439088, DOI 10.1093/imrn/rnu082
Additional Information
- Martin Meurer
- Affiliation: Institut für Mathematik, RWTH Aachen University, Templergraben 55, D-52062 Aachen, Germany
- Email: meurer@instmath.rwth-aachen.de
- Received by editor(s): November 16, 2015
- Received by editor(s) in revised form: May 22, 2016, and June 3, 2016
- Published electronically: August 15, 2017
- © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 1185-1250
- MSC (2010): Primary 28A75; Secondary 28A80, 42B20
- DOI: https://doi.org/10.1090/tran/7011
- MathSciNet review: 3729499