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Low-energy $α$-harmonic maps into the round sphere

Ben Sharp

Abstract

We classify low-energy $α$-harmonic maps from a closed non-spherical Riemannian surface $Σ$ of constant curvature to the round sphere via their bubble scales and centres. In particular we show that as $1<α\downarrow 1$ and assuming $E_α$ is close to $| Σ|+4π$ then degree-one $α$-harmonic maps blow a bubble based at a critical point $a_c$ of a an explicit function $\mathcal{J}$ and at scale $\sqrt{ |\mathcal{J}(a_c)|^{-1}(α-1)}$. Up to a constant, $\mathcal{J}$ is the sum of the squares of any $L^2$-orthonormal basis of holomorphic one-forms on the domain.

Low-energy $α$-harmonic maps into the round sphere

Abstract

We classify low-energy -harmonic maps from a closed non-spherical Riemannian surface of constant curvature to the round sphere via their bubble scales and centres. In particular we show that as and assuming is close to then degree-one -harmonic maps blow a bubble based at a critical point of a an explicit function and at scale . Up to a constant, is the sum of the squares of any -orthonormal basis of holomorphic one-forms on the domain.
Paper Structure (24 sections, 19 theorems, 234 equations)

This paper contains 24 sections, 19 theorems, 234 equations.

Table of Contents

  1. Introduction
  2. Outline of the paper
  3. Preliminaries
  4. Definition of the set of adapted bubbles
  5. Analytical set up and further background results
  6. Weighted inner product:
  7. Claim:
  8. Proof of the main theorems assuming further technical details
  9. Energy expansions when restricted to $\mathcal{Z}$
  10. Dealing with ${\rm I}$:
  11. Dealing with ${\rm II}$:
  12. Dealing with ${\rm III}$:
  13. Dealing with ${\rm IV}$:
  14. Estimates for ${\rm d} E_{\alpha}$ and ${\rm d}^2 E_{\alpha}$ close to $\mathcal{Z}$
  15. Estimates on $\mathcal{Z}$
  16. ...and 9 more sections

Key Result

Theorem 1.3

Let $(\Sigma, g)$ be as in Definition defSig. There exist ${\alpha}_0={\alpha}_0(\Sigma)>1\,,{\delta}={\delta}(\Sigma) >0\,,C=C(\Sigma)<\infty$ so that if $1< {\alpha} \leq {\alpha}_0$ and $u:\Sigma \to \mathbb{S}^2$ is ${\alpha}$-harmonic with $E_{\alpha}(u)\leq 4\pi + |\Sigma| + {\delta}$ then eit

Theorems & Definitions (42)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Corollary 1.5
  • Remark 1.6
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4: Notation
  • ...and 32 more