Sharp quantitative rigidity results for maps from $S^2$ to $S^2$ of general degree

TL;DR

This paper establishes sharp quantitative rigidity results for maps from the 2-sphere to itself, showing that maps with small energy defect are close to a collection of rational maps, despite the general degree case being more complex.

Contribution

The paper proves a novel quantitative rigidity estimate for maps from $S^2$ to $S^2$ with small energy defect, involving a collection of rational maps at different scales.

Findings

Maps with small energy defect are close to a collection of rational maps.

The distance is controlled by a sharp rigidity estimate involving the defect and a logarithmic term.

The result applies despite the failure of such rigidity in the general degree case.

Abstract

As the energy of any map $v$ from $S^2$ to $S^2$ is at least $4\pi \vert deg(v)\vert$ with equality if and only if $v$ is a rational map one might ask whether maps with small energy defect $\delta_v=E(v)-4\pi \vert deg(v)\vert$ are necessarily close to a rational map. While such a rigidity statement turns out to be false for maps of general degree, we will prove that any map $v$ with small energy defect is essentially given by a collection of rational maps that describe the behaviour of $v$ at very different scales and that the corresponding distance is controlled by a quantitative rigidity estimate of the form $dist^2\leq C \delta_v(1+\vert\log\delta_v\vert)$ which is indeed sharp.