Energy identity and no neck property for $\varepsilon$-harmonic and $α$-harmonic maps into homogeneous target manifolds

TL;DR

This work establishes energy identity and no-neck properties for sequences of $\varepsilon$- and $\alpha$-harmonic maps with homogeneous target manifolds. For $\varepsilon$-harmonic maps, an equivariant embedding into Euclidean space provides a Noether conservation law that underpins neck analysis and the energy identity, situating the results within the established bubbling framework. For $\alpha$-harmonic maps, a conservation-law-based approach—interpreted intrinsically—yields the same conclusions in the homogeneous target setting, extending Li–Zhu’s and Sacks–Uhlenbeck’s bubbling theory to homogeneous spaces. The paper also presents a counterexample showing that energy identity can fail in certain homotopy classes, highlighting the delicate role of topology in the bubbling process. Overall, the results broaden the applicability of neck and energy-defect analyses to homogeneous targets and offer a cohesive conservation-law framework for both regularisations of harmonic maps.

Abstract

In this paper we show the energy identity and the no-neck property for $\varepsilon$- and $α$-harmonic maps with homogeneous target manifolds. To prove this in the $\varepsilon$-harmonic case we introduce the idea of using an equivariant embedding of the homogeneous target manifold.

Theorems & Definitions (22)

• Theorem 1.1
• Theorem 1.2: Conservation law for $\varepsilon$-harmonic maps
• Remark 1.3
• Theorem 1.4: Bubbling for $\varepsilon$-harmonic maps
• Theorem 1.5: Energy identity and no neck property for $\varepsilon$-harmonic maps
• Theorem 1.6: Bubbling for $\alpha$-harmonic maps
• Theorem 1.7: Energy identity and no neck property for $\alpha$-harmonic maps
• Lemma 2.1
• proof
• proof : Proof of Theorem \ref{['theorem: conservation law']}