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Smoothness of conformal heat flow of harmonic maps

Woongbae Park

TL;DR

The paper studies the conformal heat flow for harmonic maps, coupling map evolution with a conformal metric factor on a Riemann surface. It proves there are no finite-time singularities and establishes long-time convergence: for small initial energy $E(0)$ the map collapses to a point while the conformal factor decays, and in general, under uniform energy concentration, a subsequence $t_n\to\infty$ yields convergence to a harmonic map on compact subsets away from a finite bubble set. The analysis combines local and global energy estimates, higher-regularity arguments, and bubble analysis to extend Struwe-type results to CHF, showing stability and regularity beyond previous weak-solution results. The results have implications for understanding energy concentration dynamics and global regularity in coupled geometric flows where the domain metric evolves in a conformal direction.

Abstract

The conformal heat flow of harmonic maps is a system of evolution equations combined with harmonic map flow with metric evolution in conformal direction. It is known that global weak solution of the flow exists and smooth except at mostly finitely many singular points. In this paper, we show that no finite time singularity occurs, unlike the usual harmonic map flow. And if the initial energy is small, we can obtain the uniform convergence of the map to a point and the conformal factor of the metric under some time sequence $t_n \to \infty$. Also, under the assumption that energy concentration is uniform in time, we show that there exists a sequence of time $t_n \to \infty$ such that $f(\cdot,t_n)$ converges to a harmonic map in $W^{1,2}$ on any compact set away from at most finitely many points.

Smoothness of conformal heat flow of harmonic maps

TL;DR

The paper studies the conformal heat flow for harmonic maps, coupling map evolution with a conformal metric factor on a Riemann surface. It proves there are no finite-time singularities and establishes long-time convergence: for small initial energy the map collapses to a point while the conformal factor decays, and in general, under uniform energy concentration, a subsequence yields convergence to a harmonic map on compact subsets away from a finite bubble set. The analysis combines local and global energy estimates, higher-regularity arguments, and bubble analysis to extend Struwe-type results to CHF, showing stability and regularity beyond previous weak-solution results. The results have implications for understanding energy concentration dynamics and global regularity in coupled geometric flows where the domain metric evolves in a conformal direction.

Abstract

The conformal heat flow of harmonic maps is a system of evolution equations combined with harmonic map flow with metric evolution in conformal direction. It is known that global weak solution of the flow exists and smooth except at mostly finitely many singular points. In this paper, we show that no finite time singularity occurs, unlike the usual harmonic map flow. And if the initial energy is small, we can obtain the uniform convergence of the map to a point and the conformal factor of the metric under some time sequence . Also, under the assumption that energy concentration is uniform in time, we show that there exists a sequence of time such that converges to a harmonic map in on any compact set away from at most finitely many points.
Paper Structure (6 sections, 20 theorems, 87 equations)

This paper contains 6 sections, 20 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Local and global estimate
  3. Proof of \ref{['main1']}
  4. Infinite time analysis
  5. Small energy case - Proof of \ref{['main2']}
  6. Large energy case - Proof of \ref{['main3']}

Key Result

Theorem 1.1

There exists a weak solution $f \in W^{1,2}_{loc}(M \times [0,\infty),N)$, $u \in W^{1,2}_{loc}(M \times [0,\infty))$ of eq1 that is smooth in $M \times [0,\infty)$ except at most finitely many points.

Theorems & Definitions (31)

  • Theorem 1.1: Park P23
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1: Park P23, Lemma 2
  • Lemma 2.2: Parker P96 or Sacks-Uhlenbeck SU81
  • Lemma 2.3: Park P23, Proposition 14
  • Proposition 2.4: Park P23, Lemma 25
  • Lemma 2.5
  • proof
  • ...and 21 more