2014, 21: 120-125. doi: 10.3934/era.2014.21.120

Number of extremal subsets in Alexandrov spaces and rigidity

1. 

St. Petersburg Department of V.A. Steklov Mathematical Institute, Fontanka 27, 191023, St. Petersburg

Received  April 2014 Published  July 2014

In this paper we announce the following result. We show that any $n$-dimensional nonnegatively curved Alexandrov space with the maximal possible number of extremal points is isometric to a quotient space of $\mathbb{R}^n$ by an action of a crystallographic group. We describe all such actions. We start with a history, results and open questions concerning estimates on the number of extremal subsets in Alexandrov spaces. Then we give the plan of the proof of our result; the complete proof will published elsewhere.
Citation: Nina Lebedeva. Number of extremal subsets in Alexandrov spaces and rigidity. Electronic Research Announcements, 2014, 21: 120-125. doi: 10.3934/era.2014.21.120
References:
[1]

E. Ackerman and O. Ben-Zwib, On sets of points that determine only acute angles, European J. Combin., 30 (2009), 908-910. doi: 10.1016/j.ejc.2008.07.020.

[2]

S. Alexander and R. Bishop, A cone splitting theorem for Alexandrov spaces, Pacific Journal of Mathematics, 218 (2005), 1-16.

[3]

S. Alexander, V. Kapovitch and A. Petrunin, Alexandrov Geometry. Available from: https://www.math.psu.edu/petrunin/papers/alexandrov-geometry/.

[4]

S. Alexander, V. Kapovitch and A. Petrunin, Alexandrov meets Kirszbraun, in Proceedings of the Gökova Geometry-Topology Conference 2010 (eds. S. Akbulut, D. Auroux and T. Onder), International Press, Somerville, MA, 2011, 88-109.

[5]

D. Bevan, Sets of points determining only acute angles and some related colouring problems, Electron. J. Combin., 13 (2006), Research Paper 12, 24 pp. (electronic).

[6]

L. V. Buchok, Two new approaches to obtaining estimates in the Danzer-Grünbaum problem, Math. Notes, 87 (2010), 489-496. doi: 10.1134/S0001434610030272.

[7]

Yu. Burago, M. Gromov and G. Perel'man, A. D. Aleksandrov spaces with curvatures bounded below, Uspekhi Mat. Nauk, 47 (1992), 3-51, 222; translation in Russian Math. Surveys, 47 (1992), 1-58. doi: 10.1070/RM1992v047n02ABEH000877.

[8]

L. Danzer and B. Grünbaun, Über zwei Probleme bezüglich konvexer Körper von P. Erdős und von V. L. Klee, (German) Math. Z., 79 (1962), 95-99. doi: 10.1007/BF01193107.

[9]

P. Erdős and Z. Fűredi, The greatest angle among n points in the d-dimensional Euclidean space, in Combinatorial Mathematics (Marseille-Luminy, 1981), North-Holland Math. Stud., 75, North-Holland, Amsterdam, 1983, 275-283.

[10]

P. Erdős, Some unsolved problems, Michigan Math. J., 4 (1957), 291-300. doi: 10.1307/mmj/1028997963.

[11]

K. Grove and P. Petersen, A radius sphere theorem, Invent. Math., 112 (1993), 577-583. doi: 10.1007/BF01232447.

[12]

, V. Kapovich, Private conversation.

[13]

U. Lang and V. Shroeder, Kirszbraun's theorem and metric spaces of bounded curvature, Geom. Funct. Anal., 7 (1997), 535-560. doi: 10.1007/s000390050018.

[14]

N. Lebedeva, Number of subgroups in a Bieberbach group. Available from: http://mathoverflow.net/questions/13714.

[15]

N. Lebedeva and A. Petrunin, Local characterization of polyhedral spaces, preprint, arXiv:1402.6670.

[16]

N. Li, Volume and gluing rigidity in Alexandrov geometry. Available from: http://front.math.ucdavis.edu/1110.5498.

[17]

G. Ya. Perel'man, Elements of Morse theory on Aleksandrov spaces, St. Petersbg. Math. J., 5 (1994), 205-213.

[18]

G. Ya. Perel'man and A. M. Petrunin, Extremal subsets in Aleksandrov spaces and the generalized Liberman theorem, (Russian) Algebra i Analiz, 5 (1993), 242-256; translation in St. Petersburg Math. J., 5 (1994), 215-227.

[19]

G. Ya. Perel'man and A. M. Petrunin, Quasigeodesics and gradient curves in Alexandrov spaces, preprint, 1994.

[20]

, G. Ya. Perel'man, Private conversation.

[21]

G. Ya. Perel'man, Spaces with curvature bounded below, in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), Birkhäuser, Basel, 1995, 517-525.

[22]

, A. Petrunin, Private conversation.

[23]

A. Petrunin, Parallel transportation for Alexandrov space with curvature bounded below, Geom. Funct. Anal., 8 (1998), 123-148. doi: 10.1007/s000390050050.

[24]

A. Petrunin, Semiconcave functions in Alexandrov's geometry, in Surveys in Differential Geometry, Vol. XI, Int. Press, Somerville, MA, 2007, 137-201.

[25]

A. Wörner, Boundary Strata of Nonnegatively Curved Alexandrov Spaces and a Splitting Theorem, Ph.D Thesis, Westfälischen Wilhelms-Universität Münster, 2010. Available from: https://ivv5.uni-muenster.de/u/andreas.woerner/files/diss_woerner.pdf.

show all references

References:
[1]

E. Ackerman and O. Ben-Zwib, On sets of points that determine only acute angles, European J. Combin., 30 (2009), 908-910. doi: 10.1016/j.ejc.2008.07.020.

[2]

S. Alexander and R. Bishop, A cone splitting theorem for Alexandrov spaces, Pacific Journal of Mathematics, 218 (2005), 1-16.

[3]

S. Alexander, V. Kapovitch and A. Petrunin, Alexandrov Geometry. Available from: https://www.math.psu.edu/petrunin/papers/alexandrov-geometry/.

[4]

S. Alexander, V. Kapovitch and A. Petrunin, Alexandrov meets Kirszbraun, in Proceedings of the Gökova Geometry-Topology Conference 2010 (eds. S. Akbulut, D. Auroux and T. Onder), International Press, Somerville, MA, 2011, 88-109.

[5]

D. Bevan, Sets of points determining only acute angles and some related colouring problems, Electron. J. Combin., 13 (2006), Research Paper 12, 24 pp. (electronic).

[6]

L. V. Buchok, Two new approaches to obtaining estimates in the Danzer-Grünbaum problem, Math. Notes, 87 (2010), 489-496. doi: 10.1134/S0001434610030272.

[7]

Yu. Burago, M. Gromov and G. Perel'man, A. D. Aleksandrov spaces with curvatures bounded below, Uspekhi Mat. Nauk, 47 (1992), 3-51, 222; translation in Russian Math. Surveys, 47 (1992), 1-58. doi: 10.1070/RM1992v047n02ABEH000877.

[8]

L. Danzer and B. Grünbaun, Über zwei Probleme bezüglich konvexer Körper von P. Erdős und von V. L. Klee, (German) Math. Z., 79 (1962), 95-99. doi: 10.1007/BF01193107.

[9]

P. Erdős and Z. Fűredi, The greatest angle among n points in the d-dimensional Euclidean space, in Combinatorial Mathematics (Marseille-Luminy, 1981), North-Holland Math. Stud., 75, North-Holland, Amsterdam, 1983, 275-283.

[10]

P. Erdős, Some unsolved problems, Michigan Math. J., 4 (1957), 291-300. doi: 10.1307/mmj/1028997963.

[11]

K. Grove and P. Petersen, A radius sphere theorem, Invent. Math., 112 (1993), 577-583. doi: 10.1007/BF01232447.

[12]

, V. Kapovich, Private conversation.

[13]

U. Lang and V. Shroeder, Kirszbraun's theorem and metric spaces of bounded curvature, Geom. Funct. Anal., 7 (1997), 535-560. doi: 10.1007/s000390050018.

[14]

N. Lebedeva, Number of subgroups in a Bieberbach group. Available from: http://mathoverflow.net/questions/13714.

[15]

N. Lebedeva and A. Petrunin, Local characterization of polyhedral spaces, preprint, arXiv:1402.6670.

[16]

N. Li, Volume and gluing rigidity in Alexandrov geometry. Available from: http://front.math.ucdavis.edu/1110.5498.

[17]

G. Ya. Perel'man, Elements of Morse theory on Aleksandrov spaces, St. Petersbg. Math. J., 5 (1994), 205-213.

[18]

G. Ya. Perel'man and A. M. Petrunin, Extremal subsets in Aleksandrov spaces and the generalized Liberman theorem, (Russian) Algebra i Analiz, 5 (1993), 242-256; translation in St. Petersburg Math. J., 5 (1994), 215-227.

[19]

G. Ya. Perel'man and A. M. Petrunin, Quasigeodesics and gradient curves in Alexandrov spaces, preprint, 1994.

[20]

, G. Ya. Perel'man, Private conversation.

[21]

G. Ya. Perel'man, Spaces with curvature bounded below, in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), Birkhäuser, Basel, 1995, 517-525.

[22]

, A. Petrunin, Private conversation.

[23]

A. Petrunin, Parallel transportation for Alexandrov space with curvature bounded below, Geom. Funct. Anal., 8 (1998), 123-148. doi: 10.1007/s000390050050.

[24]

A. Petrunin, Semiconcave functions in Alexandrov's geometry, in Surveys in Differential Geometry, Vol. XI, Int. Press, Somerville, MA, 2007, 137-201.

[25]

A. Wörner, Boundary Strata of Nonnegatively Curved Alexandrov Spaces and a Splitting Theorem, Ph.D Thesis, Westfälischen Wilhelms-Universität Münster, 2010. Available from: https://ivv5.uni-muenster.de/u/andreas.woerner/files/diss_woerner.pdf.

[1]

Anton Petrunin. Harmonic functions on Alexandrov spaces and their applications. Electronic Research Announcements, 2003, 9: 135-141.

[2]

Roberta Ghezzi, Frédéric Jean. A new class of $(H^k,1)$-rectifiable subsets of metric spaces. Communications on Pure and Applied Analysis, 2013, 12 (2) : 881-898. doi: 10.3934/cpaa.2013.12.881

[3]

Michel L. Lapidus, Goran Radunović, Darko Žubrinić. Fractal tube formulas and a Minkowski measurability criterion for compact subsets of Euclidean spaces. Discrete and Continuous Dynamical Systems - S, 2019, 12 (1) : 105-117. doi: 10.3934/dcdss.2019007

[4]

Changguang Dong. On density of infinite subsets I. Discrete and Continuous Dynamical Systems, 2019, 39 (5) : 2343-2359. doi: 10.3934/dcds.2019099

[5]

Kristian Bjerklöv, Russell Johnson. Minimal subsets of projective flows. Discrete and Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 493-516. doi: 10.3934/dcdsb.2008.9.493

[6]

Jérôme Bertrand. Prescription of Gauss curvature on compact hyperbolic orbifolds. Discrete and Continuous Dynamical Systems, 2014, 34 (4) : 1269-1284. doi: 10.3934/dcds.2014.34.1269

[7]

Anke D. Pohl. Symbolic dynamics for the geodesic flow on two-dimensional hyperbolic good orbifolds. Discrete and Continuous Dynamical Systems, 2014, 34 (5) : 2173-2241. doi: 10.3934/dcds.2014.34.2173

[8]

Dominique Lecomte. Hurewicz-like tests for Borel subsets of the plane. Electronic Research Announcements, 2005, 11: 95-102.

[9]

Donghi Lee, Makoto Sakuma. Simple loops on 2-bridge spheres in Heckoid orbifolds for 2-bridge links. Electronic Research Announcements, 2012, 19: 97-111. doi: 10.3934/era.2012.19.97

[10]

Xiaojun Huang, Zhiqiang Li, Yunhua Zhou. A variational principle of topological pressure on subsets for amenable group actions. Discrete and Continuous Dynamical Systems, 2020, 40 (5) : 2687-2703. doi: 10.3934/dcds.2020146

[11]

T. A. Shaposhnikova, M. N. Zubova. Homogenization problem for a parabolic variational inequality with constraints on subsets situated on the boundary of the domain. Networks and Heterogeneous Media, 2008, 3 (3) : 675-689. doi: 10.3934/nhm.2008.3.675

[12]

Motahhareh Gharahi, Shahram Khazaei. Reduced access structures with four minimal qualified subsets on six participants. Advances in Mathematics of Communications, 2018, 12 (1) : 199-214. doi: 10.3934/amc.2018014

[13]

Dou Dou, Meng Fan, Hua Qiu. Topological entropy on subsets for fixed-point free flows. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 6319-6331. doi: 10.3934/dcds.2017273

[14]

Carmen Calvo-Jurado, Juan Casado-Díaz, Manuel Luna-Laynez. The homogenization of the heat equation with mixed conditions on randomly subsets of the boundary. Conference Publications, 2013, 2013 (special) : 85-94. doi: 10.3934/proc.2013.2013.85

[15]

Olivier Goubet. Regularity of extremal solutions of a Liouville system. Discrete and Continuous Dynamical Systems - S, 2019, 12 (2) : 339-345. doi: 10.3934/dcdss.2019023

[16]

Lidong Wang, Xiang Wang, Fengchun Lei, Heng Liu. Mixing invariant extremal distributional chaos. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6533-6538. doi: 10.3934/dcds.2016082

[17]

Juan Dávila, Louis Dupaigne, Marcelo Montenegro. The extremal solution of a boundary reaction problem. Communications on Pure and Applied Analysis, 2008, 7 (4) : 795-817. doi: 10.3934/cpaa.2008.7.795

[18]

Knut Hüper, Irina Markina, Fátima Silva Leite. A Lagrangian approach to extremal curves on Stiefel manifolds. Journal of Geometric Mechanics, 2021, 13 (1) : 55-72. doi: 10.3934/jgm.2020031

[19]

Guillaume Warnault. Regularity of the extremal solution for a biharmonic problem with general nonlinearity. Communications on Pure and Applied Analysis, 2009, 8 (5) : 1709-1723. doi: 10.3934/cpaa.2009.8.1709

[20]

Alejandro Allendes, Alexander Quaas. Multiplicity results for extremal operators through bifurcation. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 51-65. doi: 10.3934/dcds.2011.29.51

2020 Impact Factor: 0.929

Metrics

  • PDF downloads (73)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]