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

Fang-Hua Lin, Antonio Segatti, Yannick Sire, Changyou Wang

TL;DR

The paper develops a novel variational elliptic regularization ( Weighted-Energy-Dissipation, $\mathcal{I}_\varepsilon$) to construct global weak solutions for the heat flow of harmonic maps into $CAT(0)$ spaces, circumventing smooth-target restrictions. It recasts the evolution as a gradient flow for a geodesically convex value function $V_\varepsilon$ and proves convergence as $\varepsilon\to 0$ to a suitable weak solution $u$, which satisfies an Evolution Variational Inequality and enjoys Hölder continuity. A parabolic Almgren–Poon type frequency function is introduced and shown to be monotone, yielding quantitative control that implies spatial Lipschitz regularity under a pinching/embeddability condition (Assumption (H)). The results extend to smooth NPC targets, where a Bochner–Harnack framework provides uniform gradient bounds and an alternative route to the Eells–Sampson theory. Overall, the work provides a robust variational approach to heat flows into singular targets and establishes sharp regularity with broad geometric applicability.

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 a wide class of CAT$(0)$ spaces, 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.

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

TL;DR

The paper develops a novel variational elliptic regularization ( Weighted-Energy-Dissipation, ) to construct global weak solutions for the heat flow of harmonic maps into spaces, circumventing smooth-target restrictions. It recasts the evolution as a gradient flow for a geodesically convex value function and proves convergence as to a suitable weak solution , which satisfies an Evolution Variational Inequality and enjoys Hölder continuity. A parabolic Almgren–Poon type frequency function is introduced and shown to be monotone, yielding quantitative control that implies spatial Lipschitz regularity under a pinching/embeddability condition (Assumption (H)). The results extend to smooth NPC targets, where a Bochner–Harnack framework provides uniform gradient bounds and an alternative route to the Eells–Sampson theory. Overall, the work provides a robust variational approach to heat flows into singular targets and establishes sharp regularity with broad geometric applicability.

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 a wide class of CAT spaces, 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.
Paper Structure (20 sections, 29 theorems, 285 equations, 1 figure)

This paper contains 20 sections, 29 theorems, 285 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Weighted Energy Dissipation functional for Harmonic maps
  3. The Convergence proof for CAT(0) targets
  4. The Value Function and the Dynamic Programming Principle
  5. Monotonicity of parabolic frequency functions
  6. $\mathcal{L}_\varepsilon$-subharmonicity of $e_\varepsilon(u_\varepsilon)$
  7. Stationarity identity of $u_\varepsilon$
  8. Energy bounds of $u_\varepsilon$
  9. Energy monotonicity of $u_\varepsilon$
  10. Monotonicity of frequency functions for $u$
  11. Regularity of the limiting map $u$
  12. Hölder continuity
  13. Lower bound of frequency function
  14. Proof Theorem \ref{['Lip_reg']}
  15. WED-approximation for smooth NPC target manifolds
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $(X,d)$ be a CAT(0)-space and ${u}_0\in H^1(M,X)$. Then

Figures (1)

  • Figure 1: The homogeneous tree with $Q=6$

Theorems & Definitions (60)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.1: Existence of a Gradient Flow and Hölderianity of solutions
  • Remark 1.1
  • Theorem 1.2: Spatial Lipschitz regularity of solutions
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.2
  • Remark 1.3
  • Lemma 2.1: Convexity of $E$
  • ...and 50 more