Discrete harmonic maps between hyperbolic surfaces

2024/05/03 by Lam, Wai Yeung · 3 citations
#Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2405.02205

Abstract

Given a topological cell decomposition of a closed surface equipped with edge weights, we consider the Dirichlet energy of any geodesic realization of the 1-skeleton graph to a hyperbolic surface. By minimizing the energy over all possible hyperbolic structures and over all realizations within a fixed homotopy class, one obtains a discrete harmonic map into an optimal hyperbolic surface. We characterize the extremum by showing that at the optimal hyperbolic structure, the discrete harmonic map and the edge weights are induced from a weighted Delaunay decomposition.

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• A Structure-Preserving Numerical Method for Harmonic Maps Between High-genus Surfaces
• Period matrices and homological quasi-trees on discrete Riemann surfaces
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