# A gap theorem for $\alpha$-harmonic maps between two-spheres

## TL;DR

This paper establishes an optimal gap theorem for $$-harmonic maps between two-spheres, analyzing limits of these maps with bounded energy using recent energy identities, and extends previous work on harmonic map approximations.

## Contribution

It provides a new optimal gap theorem for $$-harmonic maps of degree -1, 0, or 1, advancing understanding of their limiting behavior and energy identities.

## Key findings

- Established an optimal gap theorem for degree -1, 0, 1 maps
- Analyzed limits of $$-harmonic maps with bounded energy
- Utilized recent energy identity to derive results

## Abstract

In this paper we consider approximations introduced by Sacks-Uhlenbeck of the harmonic energy for maps from $S^2$ into $S^2$. We continue the analysis in [6] about limits of $\alpha$-harmonic maps with uniformly bounded energy. Using a recent energy identity in [7], we obtain an optimal gap theorem for the $\alpha$-harmonic maps of degree $-1, 0$ or $1$.