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

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