In this work, we propose an effective way to compute the Teichm¨uller extremal maps between Riemann
surfaces. The goal is to develop a consistent and effective algorithm for surface registration, which matches
corresponding landmark features. Surfaces are cut open along landmark curves and become multiply-connected. According to Teichmuller Quasiconformal geometry, there is an unique Teichm¨uller extremal map between two multiply-connected surfaces that minimizes the conformality distortion as much as possible. Given two multiplyconnected domains, we consider their quadratic differentials which can be expressed explicitly in closed forms. By comparing the horizontal and vertical trajectories, we can find the corresponding quadratic differentials associated uniquely to the Teichmuller extremal map. The resulting extremal map can then be computed by solving a Laplace equation on a unit disk with a given boundary condition. This method can be easily applied for surface registration between open surfaces with fixed landmark points on the boundary. We have implemented our algorithm and experimented it on both synthetic and real data. Experimental results show that our algorithm can effectively compute the Teichmuller extremal map and is useful in surface registration for shape analysis.