Singular gradient flow of the distance function and homotopy equivalence

TL;DR

This paper investigates how the topology of a domain can be understood through the properties of its distance function, showing that the singular set of this function shares the same homotopy type as the domain.

Contribution

It introduces an approach based on the invariance of the singular set under gradient flow, establishing a homotopy equivalence with the domain.

Findings

The singular set of the distance function has the same homotopy type as the domain.

Gradient flow invariance preserves topological features of the singular set.

The approach confirms the encoding of topological information in the distance function.

Abstract

It is a generally shared opinion that significant information about the topology of a bounded domain $\Omega $ of a riemannian manifold $M$ is encoded into the properties of the distance, $d_{\partial\Omega}$, %, $d:\Omega\rightarrow [0,\infty [$, from the boundary of $\Omega$. To confirm such an idea we propose an approach based on the invariance of the singular set of the distance function with respect to the generalized gradient flow of of $d_{\partial\Omega}$. As an application, we deduce that such a singular set has the same homotopy type as $\Omega$.