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Lipschitz regularity of harmonic map heat flows into $CAT(0)$ spaces

Hui-Chun Zhang, Xi-Ping Zhu

Abstract

In 1964, Eells and Sampson proved the famous long-time existence and convergence for the harmonic map heat flow into non-positively curved Riemannian manifolds. Subsequently, Hamilton investigated the corresponding initial-boundary problem. In 1992, Gromov and Schoen developed a variational theory of harmonic maps into $CAT(0)$ metric spaces. This progress naturally motivated the study of the harmonic map heat flow into singular metric spaces. In the 1990s, Mayer and Jost independently studied convex functionals on $CAT(0)$ spaces and extended Crandall-Liggett's theory of gradient flows from Banach spaces to $CAT(0)$ spaces to obtain the weak solutions, called semi-group weak solutions, for the harmonic map heat flow into $CAT(0)$ spaces. Very recently, Lin, Segatti, Sire, and Wang used an elliptic approximation method to obtain another class of weak solutions, called suitable weak solutions in the sense of the Evolution Variational Inequality (EVI), to the harmonic map heat flow into $CAT(0)$ spaces. They proved that these solutions are Lipschitz in space and $\frac{1}{2}$-Hölder continuous in time. Since the semi-group weak solutions of the harmonic map heat flow enjoy the favorable long-time existence, uniqueness and well established long-time behaviors, it is natural to ask if the semi-group solutions possess the Lipschitz regularity. In the present paper, we answer this question. We show that the semi-group weak solutions of the harmonic map heat flow into CAT(0) spaces are Lipschitz continuous in both space and time. We also establish an Eells-Sampson-type Bochner inequality.

Lipschitz regularity of harmonic map heat flows into $CAT(0)$ spaces

Abstract

In 1964, Eells and Sampson proved the famous long-time existence and convergence for the harmonic map heat flow into non-positively curved Riemannian manifolds. Subsequently, Hamilton investigated the corresponding initial-boundary problem. In 1992, Gromov and Schoen developed a variational theory of harmonic maps into metric spaces. This progress naturally motivated the study of the harmonic map heat flow into singular metric spaces. In the 1990s, Mayer and Jost independently studied convex functionals on spaces and extended Crandall-Liggett's theory of gradient flows from Banach spaces to spaces to obtain the weak solutions, called semi-group weak solutions, for the harmonic map heat flow into spaces. Very recently, Lin, Segatti, Sire, and Wang used an elliptic approximation method to obtain another class of weak solutions, called suitable weak solutions in the sense of the Evolution Variational Inequality (EVI), to the harmonic map heat flow into spaces. They proved that these solutions are Lipschitz in space and -Hölder continuous in time. Since the semi-group weak solutions of the harmonic map heat flow enjoy the favorable long-time existence, uniqueness and well established long-time behaviors, it is natural to ask if the semi-group solutions possess the Lipschitz regularity. In the present paper, we answer this question. We show that the semi-group weak solutions of the harmonic map heat flow into CAT(0) spaces are Lipschitz continuous in both space and time. We also establish an Eells-Sampson-type Bochner inequality.
Paper Structure (17 sections, 37 theorems, 285 equations)

This paper contains 17 sections, 37 theorems, 285 equations.

Table of Contents

  1. Introduction
  2. Harmonic map heat flows into $CAT(0)$ spaces
  3. Main result on the Lipschitz regularity
  4. Outline of the proof and the organization of this paper
  5. Preliminaries and notations
  6. Sobolev spaces of maps
  7. $CAT(0)$ spaces
  8. Harmonic map heat flows into $CAT(0)$ spaces
  9. The subsolution property of the distance between harmonic map heat flows
  10. Local Hölder Continuity
  11. Lipschitz continuity in time
  12. Asymptotic mean value inequality for heat equations
  13. Lipschitz continuity in space-time
  14. Nonlinear Hamilton-Jacobi flows
  15. Lipschitz continuity in space variables
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $M, N$ be two compact Riemannian manifolds without boundaries. Suppose the sectional curvature of $N$ is non-positive. Then for any $u_0\in C^\infty(M,N)$, the flow (equ-1.1) admits a unique, smooth solution $u\in C^\infty(M\times[0,+\infty),N)$. Moreover, there exists a subsequence of $\{u(\cdo

Theorems & Definitions (79)

  • Theorem 1.1: Eells-Sampson ES64
  • Theorem 1.2: Lin-Segatti-Sire-Wang LSSW25+
  • Theorem 1.3
  • Remark 1.4
  • Proposition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 69 more