Abstract
We study the existence of harmonic maps and Dirac-harmonic maps from degenerating surfaces to a nonpositive 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.
Highlights
The fundamental paper [15] by Sacks and Uhlenbeck approached the theory of harmonic maps from a Riemann surface M into a Riemannian manifold N, that is critical points u : M → N of the energy functional
The difficult part consists in controlling the limit α → 1. By studying this limit behavior of a sequence of α-harmonic maps as α 1, they obtained the existence of harmonic maps and insight into the formation of bubbles
Motivated by the supersymmetric nonlinear sigma model from quantum field theory, see [7], Dirac-harmonic maps from spin Riemann surfaces into Riemannian manifolds were introduced in [4]
Summary
The fundamental paper [15] by Sacks and Uhlenbeck approached the theory of harmonic maps from a Riemann surface M into a Riemannian manifold N , that is critical points u : M → N of the energy functional. Theorem 1.1 (Compactness and energy identity) Let (Mn, hn, cn, Sn) be a sequence of closed hyperbolic surfaces of genus g > 1 degenerating to a hyperbolic Riemann surface (M, h, c, S) by collapsing finitely many pairwise disjoint simple closed geodesics {γnj , j ∈ J }. Theorem 1.3 (Existence of Dirac-harmonic maps from degenerating surfaces) Let (Mn, hn, cn, Sn) be a sequence of closed hyperbolic surfaces of genus g > 1 degenerating to a hyperbolic Riemann surface (M, h, c, S) by collapsing finitely many pairwise disjoint simple closed geodesics {γnj , j ∈ J }. Corollary 1.6 Let (Mn, hn, cn, Sn) be a sequence of closed hyperbolic surfaces of genus g > 1 degenerating to a hyperbolic Riemann surface (M, h, c, S) with only Neveu-Schwarz type punctures by collapsing finitely many pairwisely disjoint simple closed geodesics {γnj , j ∈ J }.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
