Heat flow of harmonic maps into CAT($0$)-spaces

TL;DR

This paper develops a new approach to establish global existence, uniqueness, and Lipschitz regularity of heat flow solutions for harmonic maps into CAT(0) spaces, using novel variational and monotonicity methods.

Contribution

It introduces a new method combining elliptic regularization, a dynamical variational principle, and monotonicity techniques to analyze heat flow into singular CAT(0) spaces.

Findings

Proved global existence and uniqueness of weak solutions.

Established Lipschitz continuity in spatial variables for solutions.

Provided a new proof of the Eells-Sampson theorem using these methods.

Abstract

We introduce a new approach to prove the global existence and uniqueness of suitable weak solutions of the heat flow of harmonic mappings into CAT(0) metric spaces. Our method allows also to prove Lipschitz continuity in spatial variables for such solutions into any CAT$(0)$-space, answering a long-standing open problem in the field. Our approach is based on an elliptic regularization of the gradient flow of the Dirichlet energy and even in the case of smooth Riemannian targets provides a novel viewpoint, together with a new Dynamical Variational Principle and a new proof of the celebrated Eells-Sampson theorem. The spatial Lipschitz regularity for such weak solutions is achieved by fully exploiting the variational structure of the problem at the regularized level and introducing a parabolic frequency function of Almgren-Poon type. Our contribution is the first instance of the use of monotonicity methods for parabolic deformations of maps into singular targets.