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<p>Given a topological triangulation of a closed surface equipped with edge weights, Dirichlet energy is associated to every geodesic realization of the 1-skeleton graph to a hyperbolic surface. By minimizing the energy over all realizations within a fixed homotopy class and over all possible hyperbolic structures, one obtains a discrete harmonic map into an optimal hyperbolic surface. We characterize the minimizer by studying the Weil-Petersson symplectic gradient of the energy over the Teichmüller space. We show that at the optimal hyperbolic structure, the discrete harmonic map yields an equivariant convex polyhedral surface in Minkowski space, which corresponds to a weighted Delaunay triangulation of the hyperbolic surface. Furthermore, the space of weighted Delaunay triangulations is parametrized by the edge weights.</p>

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Discrete harmonic maps between hyperbolic surfaces

  • 期刊论文

  • 发表日期：

    2025年9月8日

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  • Wai Yeung Lam

摘要

Given a topological triangulation of a closed surface equipped with edge weights, Dirichlet energy is associated to every geodesic realization of the 1-skeleton graph to a hyperbolic surface. By minimizing the energy over all realizations within a fixed homotopy class and over all possible hyperbolic structures, one obtains a discrete harmonic map into an optimal hyperbolic surface. We characterize the minimizer by studying the Weil-Petersson symplectic gradient of the energy over the Teichmüller space. We show that at the optimal hyperbolic structure, the discrete harmonic map yields an equivariant convex polyhedral surface in Minkowski space, which corresponds to a weighted Delaunay triangulation of the hyperbolic surface. Furthermore, the space of weighted Delaunay triangulations is parametrized by the edge weights.