# Centers of disks in Riemannian manifolds

## TL;DR

This paper establishes the existence of continuous centers within embedded disks in Riemannian manifolds, highlighting conditions for equivariance and limitations in higher dimensions.

## Contribution

It proves the existence of centers for embedded disks in Riemannian manifolds and characterizes when equivariant centers can be constructed.

## Key findings

- Centers exist for certain low-dimensional disks in Riemannian manifolds.
- Equivariance of centers is possible for disks of dimension up to 3, and in some cases for dimension 4.
- Counterexamples show non-existence of equivariant centers for higher dimensions.

## 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.