Integral Menger curvature and rectifiability of 𝑛-dimensional Borel sets in Euclidean 𝑁-space

Meurer, Martin

doi:10.1090/tran/7011

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