Centers of disks in Riemannian manifolds

Igor Belegradek and Mohammad Ghomi

Vol. 304 (2020), No. 2, 401–418

DOI: 10.2140/pjm.2020.304.401

Abstract

We prove the existence of a center, or continuous selection of a point, in the relative interior of $C^1$ embedded $k$-disks in Riemannian $n$-manifolds. If $k\le 3$ the center can be made equivariant with respect to the isometries of the manifold, and under mild assumptions the same holds for $k=4=n$. By contrast, for every $n\ge k\ge 6$ there are examples where an equivariant center does not exist. The center can be chosen to agree with any of the classical centers defined on the set of convex compacta in the Euclidean space.

Keywords

continuous selection, equivariant, proper actions, actions on disks

Mathematical Subject Classification 2010

Primary: 53C40, 54C65

Secondary: 52A20, 57S25

Milestones

Received: 21 May 2018

Revised: 1 July 2019

Accepted: 12 July 2019

Published: 12 February 2020