The generalised energy identity and length of necks for $\varepsilon$-harmonic maps

TL;DR

This paper extends energy identity and neck length results from $eta$-harmonic maps to $ ext{ extgreek extvarepsilon}$-harmonic maps, identifying key quantities that determine energy behavior and neck formation.

Contribution

It introduces analogues for $ ext{ extgreek extvarepsilon}$-harmonic maps of known results on energy identity and geodesic necks, with explicit quantities governing these phenomena.

Findings

Existence of quantities depending on $ ext{ extgreek extvarepsilon}$ and bubbling radius that determine energy identity and neck formation.

Calculation of energy loss and neck length based on these quantities and the biharmonic energy of the bubble.

Extension of known results from $ ext{ extgreek extvarepsilon}$-harmonic maps to a more general setting.

Abstract

In this paper we find analogues for $\varepsilon$-harmonic maps to the generalised energy identity and the existence of geodesic necks result discovered by Yuxiang Li and Youde Wang for $\alpha$-harmonic maps. In particular there exist specific quantities depending only on $\varepsilon$ and the bubbling radius which entirely determine if the full energy identity holds and if a neck forms. In the case these fail we can calculate the energy lost and the length of the geodesic neck based on only these quantities and the biharmonic energy of the bubble.