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Low energy $\varepsilon$-harmonic maps into the round sphere

Andrew M. Roberts

TL;DR

The work provides a rigorous, quantitative analysis of low-energy epsilon-harmonic maps from genus gamma >= 1 surfaces into S^2, showing that degree +/-1 maps exhibit bubble convergence to a single bubble whose location is a critical point of the Green-function-like functional J. By building an explicit finite-dimensional bubble model Z and deriving precise energy expansions and derivative bounds, the authors establish sharp relations between the bubble scale lambda and the small parameter epsilon, and prove gradient estimates for J(a) that force bubbling to occur at favorable points. The results yield a complete bubbling picture in the small-energy regime, with a bubbling rate proportional to epsilon^{1/4} and a no-neck phenomenon, and they place the epsilon-harmonic setting on par with analogous alpha- and Ginzburg-Landau analyses through a robust variational framework. This advances understanding of how approximate harmonic maps into S^2 behave on curved domains and provides a constructive approach to quantify bubble formation.

Abstract

In this paper we classify the low energy $\varepsilon$-harmonic maps from the surfaces of constant curvature with positive genus into the round sphere. We find that all such maps with degree $\pm1$ are all quantitively close to a bubble configuration with bubbles forming at special points on the domain with bubbling radius proportional to $\varepsilon^{1/4}$.

Low energy $\varepsilon$-harmonic maps into the round sphere

TL;DR

The work provides a rigorous, quantitative analysis of low-energy epsilon-harmonic maps from genus gamma >= 1 surfaces into S^2, showing that degree +/-1 maps exhibit bubble convergence to a single bubble whose location is a critical point of the Green-function-like functional J. By building an explicit finite-dimensional bubble model Z and deriving precise energy expansions and derivative bounds, the authors establish sharp relations between the bubble scale lambda and the small parameter epsilon, and prove gradient estimates for J(a) that force bubbling to occur at favorable points. The results yield a complete bubbling picture in the small-energy regime, with a bubbling rate proportional to epsilon^{1/4} and a no-neck phenomenon, and they place the epsilon-harmonic setting on par with analogous alpha- and Ginzburg-Landau analyses through a robust variational framework. This advances understanding of how approximate harmonic maps into S^2 behave on curved domains and provides a constructive approach to quantify bubble formation.

Abstract

In this paper we classify the low energy -harmonic maps from the surfaces of constant curvature with positive genus into the round sphere. We find that all such maps with degree are all quantitively close to a bubble configuration with bubbles forming at special points on the domain with bubbling radius proportional to .
Paper Structure (14 sections, 16 theorems, 149 equations)

This paper contains 14 sections, 16 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. The space of approximate bubbles
  3. Preliminary computations
  4. Final results
  5. Energy calculations
  6. Variation calculations
  7. Estimates for $z$
  8. Difference estimates
  9. Initial bounds
  10. $\lambda$ derivatives
  11. $\nabla_A$ derivatives
  12. Terms arising from the metric
  13. Extra bounds
  14. Finishing the proof of Lemma \ref{['Lemma: Raw energy bound']}

Key Result

Theorem 1.2

Let $\Sigma$ be as in def: sigma and J def. There exist $\varepsilon_0(\Sigma),\delta_0(\Sigma),C(\Sigma)>0$ such that if $0<\varepsilon<\varepsilon_0$, $u:\Sigma\rightarrow\mathbb{S}^2$ is a $\varepsilon$-harmonic map with $E_\varepsilon[u]\leq4\pi+\delta_0$ then either

Theorems & Definitions (31)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Definition 2.1
  • Remark 2.2
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • Lemma 3.3
  • ...and 21 more