Regularizing and self-avoidance effects of integral Menger curvature
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 9 (2010) no. 1, p. 145-187

We investigate geometric curvature energies on closed curves involving integral versions of the Menger curvature. In particular, we prove geometric variants of Morrey-Sobolev and Morrey-space imbedding theorems, which may be viewed as counterparts to respective results on one-dimensional sets in the context of harmonic analysis.

Classification:  28A75,  53A04,  46E35
@article{ASNSP_2010_5_9_1_145_0,
     author = {Strzelecki, Pawe\l\ and Szuma\'nska, Marta and von der Mosel, Heiko},
     title = {Regularizing and self-avoidance effects of integral Menger curvature},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 9},
     number = {1},
     year = {2010},
     pages = {145-187},
     zbl = {1193.28007},
     mrnumber = {2668877},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2010_5_9_1_145_0}
}
Strzelecki, Paweł; Szumańska, Marta; von der Mosel, Heiko. Regularizing and self-avoidance effects of integral Menger curvature. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 9 (2010) no. 1, pp. 145-187. http://www.numdam.org/item/ASNSP_2010_5_9_1_145_0/

[1] T. Ashton, J. Cantarella, M. Piatek and E. Rawdon, Self-contact sets for 50 tightly knotted and linked tubes, arXiv:math.DG/0508248 v1 (2005).

[2] J. R. Banavar, O. Gonzalez, J. H. Maddocks and A. Maritan, Self-interactions of strands and sheets, J. Stat. Phys. 110 (2003), 35–50. | MR 1966322 | Zbl 1029.82511

[3] L. M. Blumenthal and K. Menger, “Studies in Geometry”, Freeman and co., San Francisco, CA, 1970. | MR 273492 | Zbl 0204.53401

[4] J. Cantarella, J. H. G. Fu, R. B. Kusner, J. M. Sullivan and N. C. Wrinkle, Criticality for the Gehring link problem, Geom. Topol. 10 (2006), 2055–2116. | MR 2284052 | Zbl 1129.57006

[5] J. Cantarella, R. B. Kusner and J. M. Sullivan, On the minimum ropelength of knots and links, Invent. Math. 150 (2002), 257–286. | MR 1933586 | Zbl 1036.57001

[6] J. Cantarella, M. Piatek and E. Rawdon, Visualizing the tightening of knots, In: “VIS’05: Proc. of the 16th IEEE Visualization 2005”, IEEE Computer Society, Washington, DC, 2005, 575–582.

[7] M. Carlen, B. Laurie, J. H. Maddocks and J. Smutny, Biarcs, global radius of curvature, and the computation of ideal knot shapes, In: “Physical and Numerical Models in Knot Theory”, J. A. Calvo, K. C. Millett, E. J. Rawdon, A. Stasiak (eds.) Ser. on Knots and Everything 36, World Scientific, Singapore, 2005, 75–108. | MR 2197935 | Zbl 1095.57004

[8] R. H. Crowell and R. H. Fox, “Introduction to Knot Theory”, Springer, New York, 1977. (Reprint of the 1963 original, Graduate Texts in Mathematics, Vol. 57.) | MR 445489 | Zbl 0126.39105

[9] G. David and S. Semmes, “Singular Integrals and Rectifiable Sets in n : Au-delà des graphes lipschitziens”, Astériques 193, Soc. Mathématique France, Montrouge, 1991. | Zbl 0743.49018

[10] M. H. Freedman, Z.-X. He and Z. Wang, Möbius energy of knots and unknots, Ann. of Math. 139 (1994), 1–50. | MR 1259363 | Zbl 0817.57011

[11] H. Gerlach and J. H. Maddocks, Existence of ideal knots in § 3 , in preparation.

[12] H. Gerlach and H. Von Der Mosel, What are the longest ropes on the unit sphere? Preprint Nr. 32, Institut für Mathematik, RWTH Aachen University (2009); see http://www.instmath.rwth-aachen.de/~heiko/veroeffentlichungen/longest_ropes.pdf. | MR 2807140 | Zbl 1268.49050

[13] O. Gonzalez and R. De La Llave, Existence of ideal knots, J. Knot Theory Ramifications 12 (2003), 123–133. | MR 1953628 | Zbl 1028.57008

[14] O. Gonzalez and J. H. Maddocks, Global curvature, thickness, and the ideal shape of knots, Proc. Natl. Acad. Sci. USA 96 (1999), 4769–4773. | MR 1692638 | Zbl 1057.57500

[15] 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), 29–68. | MR 1883599 | Zbl 1006.49001

[16] I. Hahlomaa, Menger curvature and Lipschitz parametrizations in metric spaces, Fund. Math. 185 (2005), 143–169. | MR 2163108 | Zbl 1077.54016

[17] I. Hahlomaa, Curvature integral and Lipschitz parametrization in 1-regular metric spaces, Ann. Acad. Sci. Fenn. Math. 32 (2007), 99–123. | MR 2297880 | Zbl 1117.28001

[18] J. C. Léger, Menger curvature and rectifiability, Ann. of Math. 149 (1999), 831–869. | MR 1709304 | Zbl 0966.28003

[19] G. Lerman and J. T. Whitehouse, High-dimensional Menger-type curvatures – Part I: Geometric multipoles and multiscale inequalities, arXiv:0805.1425v1 (2008), to appear in Rev. Mat. Iberoamericana. | MR 2848529 | Zbl 1232.28007

[20] G. Lerman and J. T. Whitehouse, High-dimensional Menger-type curvatures – Part II: d-Separation and a menagerie of curvatures, Constr. Approx. 30 (2009), 325–360. | MR 2558685 | Zbl 1222.28007

[21] Y. Lin and P. Mattila, Menger curvature and C 1 -regularity of fractals, Proc. Amer. Math. Soc. 129 (2000), 1755–1762. | MR 1814107 | Zbl 0966.28004

[22] P. Mattila, Rectifiability, analytic capacity, and singular integrals, In: “Proc. ICM”, Vol. II, Berlin 1998, Doc. Math. 1998, Extra Vol. II, 657–664 (electronic). | MR 1648114 | Zbl 0917.28003 | Zbl 0904.42013

[23] P. Mattila, Search for geometric criteria for removable sets of bounded analytic functions, Cubo 6 (2004), 113–132. | MR 2116924 | Zbl 1084.30003

[24] M. Melnikov, Analytic capacity: discrete approach and curvature of measure, Sb. Mat. 186 (1995), 827–846. | Zbl 0840.30008

[25] M. Melnikov and J. Verdera, A geometric proof of the L 2 boundedness of the Cauchy integral on Lipschitz curves, Int. Math. Res. Not. 7 (1995), 325–331. | MR 1350687 | Zbl 0923.42006

[26] K. Menger, Untersuchungen über allgemeine Metrik. Vierte Untersuchung, Zur Metrik der Kurven, Math. Ann. 103 (1930), 466–501. | JFM 56.0508.04 | MR 1512632

[27] H. Pajot, “Analytic Capacity, Rectifiability, Menger Curvature and the Cauchy Integral”, Springer Lecture Notes, Vol. 1799, Springer Berlin, Heidelberg, New York, 2002. | MR 1952175 | Zbl 1043.28002

[28] R. Schul, Ahlfors-regular curves in metric spaces, Ann. Acad. Sci. Fenn. Math. 32 (2007), 437–460. | MR 2337487 | Zbl 1122.28006

[29] F. Schuricht and H. Von Der Mosel, Global curvature for rectifiable loops, Math. Z. 243 (2003), 37–77. | MR 1953048 | Zbl 1071.53001

[30] F. Schuricht and H. Von Der Mosel, Euler-Lagrange equations for nonlinearly elastic rods with self-contact, Arch. Ration. Mech. Anal. 168 (2003), 35–82. | MR 2029004 | Zbl 1030.74029

[31] F. Schuricht and H. Von Der Mosel, Characterization of ideal knots, Calc. Var. Partial Differential Equations 19 (2004), 281–305. | MR 2033143 | Zbl 1352.58004

[32] P. Strzelecki, M. Szumańska and H. Von Der Mosel, A geometric curvature double integral of Menger type for space curves Ann. Acad. Sci. Fenn. Math. 34 (2009), 195–214. | MR 2489022 | Zbl 1188.49016

[33] P. Strzelecki and H. Von Der Mosel, On a mathematical model for thick surfaces, In: “Physical and Numerical Models in Knot Theory”, J. A. Calvo, K. C. Millett, E. J. Rawdon, A. Stasiak (eds.), Ser. on Knots and Everything 36, World Scientific, Singapore, 2005, 547–564. | MR 2197957 | Zbl 1105.53003

[34] P. Strzelecki and H. Von Der Mosel, Global curvature for surfaces and area minimization under a thickness constraint, Calc. Var. Partial Differential Equations 25 (2006), 431–467. | MR 2214619 | Zbl 1096.53003

[35] P. Strzelecki and H. Von Der Mosel, On rectifiable curves with L p -bounds on global curvature: Self-avoidance, regularity, and minimizing knots, Math. Z. 257 (2007), 107–130. | MR 2318572 | Zbl 1354.49028

[36] P. Strzelecki and H. Von Der Mosel, Integral Menger curvature for surfaces, arXiv:math.CA/0911.2095 v2 (2009). | MR 2739778 | Zbl 1211.28003

[37] J. Verdera, The L 2 boundedness of the Cauchy integral and Menger curvature, In: “Harmonic Analysis and Boundary Value Problems”, Contemp. Math. 277, AMS, Providence, RI, 2001, 139–158. | MR 1840432 | Zbl 1002.42011