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

Fanghua Lin, Changyou Wang

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.

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

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 as the target and any complete Riemanan manifold , with positive injectivity radius and bounded curvature, as the domain.
Paper Structure (5 sections, 4 theorems, 92 equations)

This paper contains 5 sections, 4 theorems, 92 equations.

Table of Contents

  1. Introduction
  2. Parabolic inequality of $|\nabla u|^2$
  3. $(M,g)$: Euclidean space with standard metric
  4. $(M,g)$: complete Riemann manifold without boundary
  5. Sub-Caloricity of $|\partial_t u|^2$

Key Result

Theorem 1.1

Let $(X,d)$ be a CAT(0)-metric space, and $u_0\in H^1(M, X)$. If $u_\varepsilon={\rm{argmin}}\{\mathcal{I}_\varepsilon(u), \ u\in \mathfrak{V}_{u_0}\}$, then there exists a unique suitable weak solution $u$ of the heat flow of harmonic map from $(M,g)$ to $(X,d)$ such that If $(X,d)$ additionally is locally compact, then $u\in C^\alpha(M\times (0,\infty))$ for some $0<\alpha<1$.

Theorems & Definitions (5)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4: Uniform Lipschitz Estimate
  • Remark 2.1