In this talk, we shall discuss the compactness of solutions to some geometric PDEs over degenerating Riemann surfaces. In particular, we investigate the asymptotic analysis and qualitative behavior for a general sequence of Sacks-Uhlenbeckα-harmonic maps from degenerating Riemann surfaces. This answers an open problem proposed by J. D. Moore, aiming at developing a partial Morse theory for closed parametrized minimal surfaces in compact Riemannian manifolds with arbitrary codimensions.