[Paper Review] Computing discrete equivariant harmonic maps

Jonah Gaster, Brice Loustau+1 more|arXiv (Cornell University)|Oct 29, 2018

3D Shape Modeling and Analysis43 references4 citations

TL;DR

This paper presents a discrete, computationally effective method for computing equivariant harmonic maps from the universal cover of a surface into nonpositively curved spaces, using strongly convex discrete energy functionals and convergence guarantees for discrete heat flow and center-of-mass methods. The key contribution is the development of Harmony, a C++ software that numerically computes and visualizes these maps with provable convergence rates and geometric fidelity.

ABSTRACT

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.

Motivation & Objective

• To develop effective, discretized numerical methods for computing equivariant harmonic maps from the universal cover of a surface into nonpositively curved target spaces.
• To establish strong convexity of the discrete energy functional on triangulated surfaces, ensuring unique minimizers and convergence of optimization schemes.
• To implement and validate numerical algorithms—specifically discrete heat flow and $ anh$-center of mass methods—on hyperbolic surfaces.
• To provide a computational framework, implemented in the Harmony software, for experimental exploration of harmonic maps and their role in Teichmüller theory and nonabelian Hodge correspondence.
• To demonstrate that high-energy harmonic maps exhibit geometric behavior consistent with theory: regions of high dilation approach ideal triangles, and contraction concentrates along geodesic laminations.

Proposed method

• Discretize the energy functional using vertex and edge weights on triangulated meshes, where edge weights generalize cotangent weights and encode conformal structure.
• Prove strong convexity of the discrete energy functional in the hyperbolic plane ($\mathbb{H}^2$) and more general nonpositively curved metric spaces, ensuring uniqueness and convergence of minimizers.
• Implement discrete heat flow as a gradient descent on the Riemannian manifold of discrete maps, with explicit convergence rate bounds derived from convexity.
• Apply generalized mean value properties to define center-of-mass methods in metric spaces, particularly using the $\cosh$-center of mass for robustness in non-Euclidean settings.
• Use iterative midpoint refinement of meshes to improve resolution and convergence accuracy, enabling high-precision computation of equivariant maps.
• Develop the Harmony software in C++ with multithreaded, real-time visualization of the flow process, supporting both discrete heat flow and center-of-mass optimization.

Experimental results

Research questions

• RQ1 Can the theory of equivariant harmonic maps be discretized in a way that preserves strong convexity and enables provable convergence of numerical schemes?
• RQ2 How do discrete heat flow and center-of-mass methods compare in convergence speed and stability when computing harmonic maps into hyperbolic surfaces?
• RQ3 To what extent can numerical computation of harmonic maps reveal geometric features such as ideal triangle formation and lamination concentration in high-energy regimes?
• RQ4 Can a software framework like Harmony enable experimental investigation of deep geometric structures such as the nonabelian Hodge correspondence and Teichmüller theory?
• RQ5 What is the role of mesh refinement and discrete conformal structure in ensuring accurate and stable numerical computation of equivariant harmonic maps?

Key findings

• The discrete energy functional on triangulated surfaces is strongly convex in nonpositively curved target spaces, including $\mathbb{H}^2$, ensuring a unique minimizer.
• The discrete heat flow converges to the energy minimizer with an explicit convergence rate, derived from the strong convexity of the energy functional.
• The $\cosh$-center of mass method provides a robust alternative to gradient descent, particularly effective in non-Euclidean settings with improved numerical stability.
• High-energy harmonic maps computed via Harmony exhibit the predicted geometric behavior: regions of maximal dilation form ideal hyperbolic triangles, and regions of maximal contraction approach a geodesic lamination.
• The software Harmony successfully visualizes the evolution of the discrete heat flow in real time, with convergence confirmed both numerically and geometrically.
• The boundary map $\partial H$ from measured foliations to measured laminations is visually confirmed by the flow: as energy increases, the image of the map concentrates along geodesic lamination structures.