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A novel method for registering 3D surfaces with large deformations is presented, which is based on quasi-conformal geometry. A general diffeomorphism distorts the conformal structure of the surface, which is represented as the Beltrami coefficient. Inversely, the diffeomorphism can be determined by the Beltrami coefficient in an essentially unique way. Our registration method first extracts the features on the surfaces, then estimates the Beltrami coefficient, and finally uniquely determines the registration mapping by solving Beltrami equations using curvature flow. The method is 1) general, it can search the desired registration in the whole space of diffeomorphisms, which includes the conventional searching spaces, such as rigid motions, isometric transformations or conformal mappings; 2) global optimal, the global optimum is determined by the method unique up to a 3 dimensional transformation group; 3) robust, it handles large surfaces with complicated topologies; 4) rigorous, it has solid theoretic foundation. Experiments on the real surfaces with large deformations and complicated topologies demonstrate the efficiency, robustness of the proposed method.