Singularities of solutions of time dependent Hamilton-Jacobi equations. Applications to Riemannian geometry

TL;DR

This paper investigates the structure of singularities in solutions to time-dependent Hamilton-Jacobi equations on manifolds, showing the singular set is locally contractible and analyzing its topological properties, with applications to Riemannian geometry.

Contribution

It proves the local contractibility of the singular set of viscosity solutions and explores its homotopy type, extending understanding of solution singularities in geometric contexts.

Findings

The singular set of solutions is locally contractible.

The homotopy type of the singular set is characterized.

Applications to the singularities of distance functions in Riemannian geometry.

Abstract

If $U:[0,+\infty[\times M$ is a uniformly continuous viscosity solution of the evolution Hamilton-Jacobi equation $$\partial_tU+ H(x,\partial_xU)=0,$$ where $M$ is a not necessarily compact manifold, and $H$ is a Tonelli Hamiltonian, we prove the set $\Sigma(U)$, of points where $U$ is not differentiable, is locally contractible. Moreover, we study the homotopy type of $\Sigma(U)$. We also give an application to the singularities of a distance function to a closed subset of a complete Riemannian manifold.