The Convergence of Discrete Uniformizations for Closed Surfaces

Abstract: The notions of discrete conformality on triangle meshes have rich mathematical theories and wide applications. The related notions of discrete uniformizations on triangle meshes, suggest efficient methods for computing the uniformizations of surfaces. This paper proves that the discrete uniformizations approximate the continuous uniformization for closed surfaces of genus $\geq1$, when the approximating triangle meshes are reasonably good. To the best of the authors' knowledge, this is the first convergence result on computing uniformizations for surfaces of genus $>1$.