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Existence of expanding harmonic map flows to hemispheres

Xuanyu Li

TL;DR

The paper addresses whether harmonic map flows starting from $0$-homogeneous data into hemispheres admit non-trivial expanding solutions and, under monotonicity constraints, whether such flows are unique. It introduces a perturbation method: approximate the initial data by $u_0^{\sigma}$ inside a regular ball, solve expanding flows into the ball, and extract a limit as $\sigma\to0$ yielding an expander into $\overline{S^m_+}$ that satisfies the parabolic monotonicity formula and parabolic stationary condition. The results establish existence of non-trivial expanders for $3\le n\le 6$ (smooth away from the origin) and, for $n\ge 7$, expanders with a codimension-$7$ singular set, thereby providing broad non-uniqueness phenomena in harmonic map flow with hemispherical targets. These findings extend prior work on expanders and energy-minimizing maps, and address Struwe's question about uniqueness under monotonicity constraints. The work has implications for understanding singularity models, compactness, and non-uniqueness in geometric evolution equations.

Abstract

We show the existence of non-trivial self-expanding harmonic map flows starting from non-energy-minimizing 0-homogeneous maps to a regular ball or a closed hemisphere. In particular, given a non-minimizing but stationary 0-homogeneous harmonic map $u_0$ to a closed hemisphere, we construct infinitely many different weak solutions to harmonic map flow starting from $u_0$, all of which satisfy the parabolic monotonicity formula. This answers a question of Struwe.

Existence of expanding harmonic map flows to hemispheres

TL;DR

The paper addresses whether harmonic map flows starting from -homogeneous data into hemispheres admit non-trivial expanding solutions and, under monotonicity constraints, whether such flows are unique. It introduces a perturbation method: approximate the initial data by inside a regular ball, solve expanding flows into the ball, and extract a limit as yielding an expander into that satisfies the parabolic monotonicity formula and parabolic stationary condition. The results establish existence of non-trivial expanders for (smooth away from the origin) and, for , expanders with a codimension- singular set, thereby providing broad non-uniqueness phenomena in harmonic map flow with hemispherical targets. These findings extend prior work on expanders and energy-minimizing maps, and address Struwe's question about uniqueness under monotonicity constraints. The work has implications for understanding singularity models, compactness, and non-uniqueness in geometric evolution equations.

Abstract

We show the existence of non-trivial self-expanding harmonic map flows starting from non-energy-minimizing 0-homogeneous maps to a regular ball or a closed hemisphere. In particular, given a non-minimizing but stationary 0-homogeneous harmonic map to a closed hemisphere, we construct infinitely many different weak solutions to harmonic map flow starting from , all of which satisfy the parabolic monotonicity formula. This answers a question of Struwe.
Paper Structure (8 sections, 9 theorems, 38 equations)

This paper contains 8 sections, 9 theorems, 38 equations.

Table of Contents

  1. introduction
  2. Related history on existence and uniqueness of harmonic map flows
  3. Sketch of proofs
  4. Organization of paper
  5. Preliminaries
  6. Harmonic map and their heat flow into regular balls
  7. Energy minimizing maps into a hemisphere
  8. Proof of main results

Key Result

Theorem 1.1

Suppose $3\leqslant n\leqslant6$. For any regular 0-homogeneous map $u_0:\mathbb{R}^n\rightarrow\overline{S^m_+}$, there exists an expanding harmonic map flow $u\in C^{\infty}(\mathbb{R}^n\times[0,\infty)\setminus\lbrace(0,0)\rbrace,\overline{S^m_+})$ starting from $u_0$.

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 1.2
  • Example
  • Theorem 1.3
  • Remark
  • Proposition 2.1
  • Proposition 2.2: Jäger-Kaul JagerKaul
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • ...and 9 more