Weak geodesics on prox-regular subsets of Riemannian manifolds

TL;DR

This paper defines and characterizes weak geodesics on prox-regular subsets of Riemannian manifolds, using a Lipschitz projection map and viscosity critical points of the energy functional.

Contribution

It introduces a new definition of weak geodesics on prox-regular sets and characterizes them via viscosity critical points, expanding geometric analysis tools.

Findings

Defined weak geodesics with weak regularities

Characterized weak geodesics as viscosity critical points

Established Lipschitz constant for the projection map

Abstract

We give a definition of weak geodesics on prox-regular subsets of Riemannian manifolds as continuous curves with some weak regularities. Then obtaining a suitable Lipschitz constant of the projection map, we characterize weak geodesics on a prox-regular set with assigned end points as viscosity critical points of the energy functional.