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

TL;DR

This paper classifies low energy epsilon-harmonic maps from constant curvature surfaces of positive genus into the sphere, showing they are close to bubble configurations with bubbles forming at specific points.

Contribution

It provides a classification of low energy epsilon-harmonic maps with degree ±1, revealing their proximity to bubble configurations and the bubbling radius dependence on epsilon.

Findings

Maps with degree ±1 are close to bubble configurations.

Bubbles form at special points with radius proportional to epsilon^{1/4}.

All low energy maps exhibit similar bubbling behavior.

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 $\varepsilon^{1/4}$.