Computing discrete equivariant harmonic maps

Jonah Gaster (1) , Brice Loustau (2, 3) , Leonard Monsaingeon (4, 5)

1 McGill University = Université McGill [Montréal, Canada]
2 TU Darmstadt - Technische Universität Darmstadt - Technical University of Darmstadt
3 NJIT - Department of Mathematical Sciences [Newark, NJ]
4 IECL - Institut Élie Cartan de Lorraine
5 GFM - Grupo de Física Matemática - Group of Mathematical Physics

We present effective methods to compute equivariant harmonic maps from the universal cover of a surface into a nonpositively curved space. By discretizing the theory appropriately, we show that the energy functional is strongly convex and derive convergence of the discrete heat flow to the energy minimizer, with explicit convergence rate. We also examine center of mass methods, after showing a generalized mean value property for harmonic maps. We feature a concrete illustration of these methods with Harmony, a computer software that we developed in C++, whose main functionality is to numerically compute and display equivariant harmonic maps.

Domaines: Géométrie différentielle [math.DG]

Jonah Gaster, Brice Loustau, Leonard Monsaingeon. Computing discrete equivariant harmonic maps. 2019. ⟨hal-02054982v1⟩