Remarks on the heat flow of harmonic maps into CAT(0)-spaces

TL;DR

This paper provides a simplified, elementary proof of the local Lipschitz regularity for heat flow harmonic maps into CAT(0)-spaces, extending previous existence results with a new approach inspired by Korevaar and Schoen.

Contribution

It introduces an alternative, elementary proof technique for regularity of harmonic map heat flow into CAT(0)-spaces, broadening understanding and simplifying prior methods.

Findings

Established local Lipschitz regularity for weak solutions

Extended applicability to any CAT(0)-space and complete Riemannian domain

Provided an elementary proof inspired by Korevaar and Schoen

Abstract

In this paper, we present an alternate, elementary proof of the local Lipschitz regularity of the suitable weak solution of heat flow of harmonic maps into CAT(0)-metric spaces, whose existence was established by Lin, Segatti, Sire, and Wang through an elliptic regularization approach. The ideas of the proof are inspired by Korevaar and Schoen, and they work for any CAT(0)-metric space $(X,d)$ as the target and any complete Riemanan manifold $(M,g)$, with positive injectivity radius and bounded curvature, as the domain.