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

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.

💡 Research Summary

The paper investigates two regularized variational problems for maps from a compact Riemannian surface (M) into a compact target manifold (N): the (\alpha)-energy introduced by Sacks–Uhlenbeck and the (\varepsilon)-energy introduced by Lamm. Both functionals satisfy the Palais–Smale condition, guaranteeing the existence of smooth critical points, called (\alpha)-harmonic and (\varepsilon)-harmonic maps respectively. The central question is whether, for a sequence of such maps whose parameters tend to the harmonic limit ((\alpha_k\to1) or (\varepsilon_k\to0)) and whose energies remain uniformly bounded, the classical bubbling phenomenon can occur without loss of energy and without formation of a “neck” region connecting the base map to the bubbles.

The authors restrict to the case where the target (N) is a homogeneous Riemannian manifold, i.e. the isometry group (G) acts transitively. A key geometric input (Theorem 1.1) is that any homogeneous manifold admits an isometric, equivariant embedding (\Phi:N\to\mathbb R^{L}) together with a group homomorphism (\Pi:G\to O(L)) such that (\Pi(\psi)\circ\Phi=\Phi\circ\psi) for every (\psi\in G). This equivariant embedding makes the (\varepsilon)-energy invariant under the induced orthogonal action, a property that fails for a generic embedding.

Using Noether’s theorem, the authors derive a conservation law for (\varepsilon)-harmonic maps (Theorem 1.2): \