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Computing harmonic maps between Riemannian manifolds

Part of: Variational problems in infinite-dimensional spaces Numerical approximation and computational geometry Global differential geometry

Published online by Cambridge University Press:  18 February 2022

Jonah Gaster ,
Brice Loustau  and
Léonard Monsaingeon
Jonah Gaster
Affiliation:
Department of Mathematical Sciences, University of Wisconsin-Milwaukee, Milwaukee, WI, USA e-mail: gaster@uwm.edu
Brice Loustau*
Affiliation:
Mathematisches Institut, Heidelberg University, Heidelberg, Germany Heidelberg Institute of Theoretical Studies (HITS), Heidelberg University, Heidelberg, Germany
Léonard Monsaingeon
Affiliation:
Institut Élie Cartan de Lorraine and Grupo de Física Matemática, IECL Université de Lorraine, Vandœuvre-lès-Nancy Cedex, France GFM Universidade de Lisboa, Lisboa, Portugal e-mail: leonard.monsaingeon@univ-lorraine.fr
*
e-mail: bloustau@mathi.uni-heidelberg.de
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Abstract

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.

Keywords

Discrete differential geometryharmonic mapsgeometric analysis

MSC classification

Primary: 58E20: Harmonic maps , etc.
Secondary: 53C43: Differential geometric aspects of harmonic maps 65D18: Computer graphics, image analysis, and computational geometry
Type
Article
Information
Canadian Journal of Mathematics , Volume 75 , Issue 2 , April 2023 , pp. 531 - 580
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Copyright
© Canadian Mathematical Society, 2022

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Footnotes

The first two authors gratefully acknowledge support from NSF Grant DMS1107367 RNMS: GEometric structures And Representation varieties (the GEAR Network). The third author was partially supported by the Portuguese Science Foundation FCT trough grant PTDC/MAT-STA/0975/2014 from Stochastic Geometric Mechanics to Mass Transportation Problems.

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