Number of extremal subsets in Alexandrov spaces and rigidity

Nina Lebedeva

doi:10.3934/era.2014.21.120

Article Contents

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

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Number of extremal subsets in Alexandrov spaces and rigidity

Nina Lebedeva^1,

1.
St. Petersburg Department of V.A. Steklov Mathematical Institute, Fontanka 27, 191023, St. Petersburg

Received: April 2014

Published online: July 2014

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Abstract

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.

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Mathematics Subject Classification: 51K10, 53C45, 20F55.

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