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Symmetric log-epiperimetric inequality for harmonic maps with analytic target and applications

Riccardo Caniato, Davide Parise

TL;DR

The paper proves a direct symmetric log-epiperimetric inequality for harmonic maps with real-analytic targets, enabling a purely variational route to uniqueness of tangents at isolated singularities and at infinity. Through a slicing lemma and Lyapunov–Schmidt reduction, it constructs a finite-dimensional reduction and provides two proofs (one via direct competitor construction and one via parabolic methods) that yield energy decay controls. As a consequence, Simon's tangent-uniqueness result is recovered variationally, and tangents at infinity are shown to be unique under suitable energy growth, with explicit logarithmic decay rates. The results deepen the connection between epiperimetric/Łojasiewicz-type inequalities and regularity, offering tools applicable to almost-minimizers and broader geometric PDE settings.

Abstract

We establish a direct symmetric (log)-epiperimetric inequality for harmonic maps with analytic target and we leverage on this result to achieve a new proof of Simon's celebrated uniqueness of tangents with isolated singularity for energy minimizing harmonic maps. Moreover, we show that tangents at infinity of energy minimizing harmonic maps with suitably controlled energy growth are always unique, by exploiting the lower bound entailed in the symmetric (log)-epiperimetric inequality.

Symmetric log-epiperimetric inequality for harmonic maps with analytic target and applications

TL;DR

The paper proves a direct symmetric log-epiperimetric inequality for harmonic maps with real-analytic targets, enabling a purely variational route to uniqueness of tangents at isolated singularities and at infinity. Through a slicing lemma and Lyapunov–Schmidt reduction, it constructs a finite-dimensional reduction and provides two proofs (one via direct competitor construction and one via parabolic methods) that yield energy decay controls. As a consequence, Simon's tangent-uniqueness result is recovered variationally, and tangents at infinity are shown to be unique under suitable energy growth, with explicit logarithmic decay rates. The results deepen the connection between epiperimetric/Łojasiewicz-type inequalities and regularity, offering tools applicable to almost-minimizers and broader geometric PDE settings.

Abstract

We establish a direct symmetric (log)-epiperimetric inequality for harmonic maps with analytic target and we leverage on this result to achieve a new proof of Simon's celebrated uniqueness of tangents with isolated singularity for energy minimizing harmonic maps. Moreover, we show that tangents at infinity of energy minimizing harmonic maps with suitably controlled energy growth are always unique, by exploiting the lower bound entailed in the symmetric (log)-epiperimetric inequality.
Paper Structure (9 sections, 12 theorems, 139 equations)

This paper contains 9 sections, 12 theorems, 139 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on harmonic maps
  3. A slicing lemma and Lyapunov--Schmidt reduction
  4. The symmetric log-epiperimetric inequality for harmonic maps (Theorem \ref{['Theorem: (log)-epiperimetric inequality for harmonic maps']})
  5. The integrable case
  6. Proof of the uniqueness of tangent maps with isolated singularities (Theorem \ref{['Theorem: uniqueness of tangent maps with isolated singularities']})
  7. Proof of the uniqueness of tangent maps at infinity (Theorem \ref{['thm: uniqueness of blow-downs']})
  8. A criterion for the uniqueness of tangent maps
  9. Epiperimetric and Ł ojasiewicz--Simon inequalities

Key Result

Theorem 1.1

Let $N\subset\mathbb{R}^k$ be a closed real-analytic submanifold in $\mathbb{R}^k$ and let $n\in\mathbb{N}$ be such that $n\ge 3$. Let $u_0\in C^{\infty}(\mathbb{S}^{n-1},N)$ be a harmonic map on $\mathbb{S}^{n-1}$ and let $\tilde{u}_0\in W^{1,2}(\mathbb{B}^n,N)$ be its 0-homogeneous extension insid There exist constant $\varepsilon, \delta > 0$, and $\gamma \in [0, 1)$ depending on the dimension

Theorems & Definitions (25)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Definition 2.1: Dirichlet energy and harmonic maps
  • Definition 2.2
  • Lemma 2.3
  • proof
  • ...and 15 more