Existence of (Dirac-)harmonic Maps from Degenerating (Spin) Surfaces

TL;DR

This paper investigates the existence and convergence of harmonic and Dirac-harmonic maps from degenerating surfaces to non-positive curved manifolds, establishing energy identities and existence results under bounded energy conditions.

Contribution

It introduces a new approach using $eta$-harmonic maps to analyze convergence and energy identities for maps from degenerating surfaces, leading to existence results for nontrivial harmonic and Dirac-harmonic maps.

Findings

Convergence and energy identity without energy loss near punctures.

Existence of nontrivial harmonic and Dirac-harmonic maps from degenerating surfaces.

Energy bounds ensure nontrivial limiting maps.

Abstract

We study the existence of harmonic maps and Dirac-harmonic maps from degenerating surfaces to non-positive curved manifold via the scheme of Sacks and Uhlenbeck. By choosing a suitable sequence of $\alpha$-(Dirac-)harmonic maps from a sequence of suitable closed surfaces degenerating to a hyperbolic surface, we get the convergence and a cleaner energy identity under the uniformly bounded energy assumption. In this energy identity, there is no energy loss near the punctures. As an application, we obtain an existence result about (Dirac-)harmonic maps from degenerating (spin) surfaces. If the energies of the map parts also stay away from zero, which is a necessary condition, both the limiting harmonic map and Dirac-harmonic map are nontrivial.