Geodesic
Computing harmonic maps between Riemannian manifolds
Canadian journal of mathematics, Tome 75 (2023) no. 2, pp. 531-580

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DOI

In our previous paper (Gaster et al., 2018, arXiv:1810.11932), we showed that the theory of harmonic maps between Riemannian manifolds, especially hyperbolic surfaces, may be discretized by introducing a triangulation of the domain manifold with independent vertex and edge weights. In the present paper, we study convergence of the discrete theory back to the smooth theory when taking finer and finer triangulations, in the general Riemannian setting. We present suitable conditions on the weighted triangulations that ensure convergence of discrete harmonic maps to smooth harmonic maps, introducing the notion of (almost) asymptotically Laplacian weights, and we offer a systematic method to construct such weighted triangulations in the two-dimensional case. Our computer software Harmony successfully implements these methods to compute equivariant harmonic maps in the hyperbolic plane.
DOI : 10.4153/S0008414X22000074
Mots-clés : Discrete differential geometry, harmonic maps, geometric analysis 
Gaster, Jonah; Loustau, Brice; Monsaingeon, Léonard. Computing harmonic maps between Riemannian manifolds. Canadian journal of mathematics, Tome 75 (2023) no. 2, pp. 531-580. doi: 10.4153/S0008414X22000074
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     title = {Computing harmonic maps between {Riemannian} manifolds},
     journal = {Canadian journal of mathematics},
     pages = {531--580},
     year = {2023},
     volume = {75},
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     doi = {10.4153/S0008414X22000074},
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