Uniform boundedness on extremal subsets in Alexandrov spaces

Abstract

In this paper, we study extremal subsets in Alexandrov spaces with dimension $n$, curvature $\ge \kappa$, and diameter $\le D$. We show that the following three quantities are uniformly bounded above in terms of $n$, $\kappa$, and $D$: (1) the number of extremal subsets in an Alexandrov space; (2) the Betti numbers of an extremal subset; (3) the volume of an extremal subset. The proof is an application of essential coverings introduced by Yamaguchi.

rewritten introduction, added references, and other minor changes; to appear in J. Geom. Anal

References in corpus (1)

• Regular points of extremal subsets in Alexandrov spaces