Relative energy gap for harmonic maps of Riemann surfaces into real analytic Riemannian manifolds

TL;DR

This paper generalizes the energy gap result for harmonic maps from Riemann surfaces into Riemannian manifolds, showing that maps with high energy close to a reference map exhibit similar gap properties.

Contribution

It extends the classical energy gap theorem to harmonic maps with high absolute energy but small relative energy compared to a reference map.

Findings

Established a relative energy gap for harmonic maps

Extended Sacks-Uhlenbeck result to high-energy maps

Provided conditions for energy gap based on relative energy

Abstract

We extend the well-known Sacks-Uhlenbeck energy gap result (1981) for harmonic maps from closed Riemann surfaces into closed Riemannian manifolds from the case of maps with small energy (thus near a constant map), to the case of harmonic maps with high absolute energy but small energy relative to a reference harmonic map.