SRA-free condition by Zolotov for self-contracted curves and nondegeneracy of the zz-distance for Möbius structures on the circle

Buyalo, S.

doi:10.1090/spmj/1624

SRA-free condition by Zolotov for self-contracted curves and nondegeneracy of the zz-distance for Möbius structures on the circle
HTML articles powered by AMS MathViewer

by S. Buyalo
Translated by: S. Buyalo

St. Petersburg Math. J. 31 (2020), 819-829

DOI: https://doi.org/10.1090/spmj/1624

Published electronically: September 3, 2020

PDF | Request permission

Abstract:

The SRA-free condition for metric spaces (that is, spaces without Small Rough Angles) was introduced by Zolotov to study rectifiability for self-contracted curves in various metric spaces. A Möbius invariant version of this notion is introduced, which allows one to show that the zz-distance associated with the respective Möbius structure on the circle is nondegenerate. This result is an important part of a solution to the inverse problem of Möbius geometry on the circle.

References

S. V. Buyalo, Möbius structures and timed causal spaces on the circle, Algebra i Analiz 29 (2017), no. 5, 1–50 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 29 (2018), no. 5, 715–747. MR 3724637, DOI 10.1090/spmj/1513
—, On the inverse problem of Möbius geometry on the circle, arXiv:math.MG/1810.03133.
A. Daniilidis, G. David, E. Durand-Cartagena, and A. Lemenant, Rectifiability of self-contracted curves in the Euclidean space and applications, J. Geom. Anal. 25 (2015), no. 2, 1211–1239. MR 3319969, DOI 10.1007/s12220-013-9464-z
Aris Daniilidis, Robert Deville, Estibalitz Durand-Cartagena, and Ludovic Rifford, Self-contracted curves in Riemannian manifolds, J. Math. Anal. Appl. 457 (2018), no. 2, 1333–1352. MR 3705356, DOI 10.1016/j.jmaa.2017.04.011
Aris Daniilidis, Olivier Ley, and Stéphane Sabourau, Asymptotic behaviour of self-contracted planar curves and gradient orbits of convex functions, J. Math. Pures Appl. (9) 94 (2010), no. 2, 183–199 (English, with English and French summaries). MR 2665417, DOI 10.1016/j.matpur.2010.03.007
Thomas Foertsch and Viktor Schroeder, Ptolemy circles and Ptolemy segments, Arch. Math. (Basel) 98 (2012), no. 6, 571–581. MR 2935663, DOI 10.1007/s00013-012-0398-7
Thomas Foertsch and Viktor Schroeder, Metric Möbius geometry and a characterization of spheres, Manuscripta Math. 140 (2013), no. 3-4, 613–620. MR 3019142, DOI 10.1007/s00229-012-0555-0
Antoine Lemenant, Rectifiability of non Euclidean planar self-contracted curves, Confluentes Math. 8 (2016), no. 2, 23–38. MR 3633219, DOI 10.5802/cml.31
Shin-ichi Ohta, Self-contracted curves in $\textrm {CAT}(0)$-spaces and their rectifiability, J. Geom. Anal. 30 (2020), no. 1, 936–967. MR 4058544, DOI 10.1007/s12220-018-00126-7
N. Lebedeva, Ohta Shin-ichi, and V. Zolotov, Self-contracted curves in spaces with weak lower curvature bound, arXiv:1902.01594.
Eugene Stepanov and Yana Teplitskaya, Self-contracted curves have finite length, J. Lond. Math. Soc. (2) 96 (2017), no. 2, 455–481. MR 3708958, DOI 10.1112/jlms.12068
V. Zolotov, Subsets with small angles in self-contraced curves, arXiv:1804.00234 (2018).