On the computation of harmonic maps by unconstrained algorithms based on totally geodesic embeddings

TL;DR

This paper introduces a general unconstrained algorithm for computing harmonic maps between Riemannian manifolds by embedding the target into Euclidean space, simplifying discretization and avoiding projections or Lagrange multipliers.

Contribution

The paper presents a novel unconstrained approach using totally geodesic embeddings to compute harmonic maps, eliminating the need for projections or Lagrange multipliers.

Findings

Error estimates established for stationary solutions

Algorithm successfully tested numerically

Avoids traditional projection-based methods

Abstract

In this paper, we present an algorithm for the computation of harmonic maps, and respectively, of the harmonic map heat flow between two closed Riemannian manifolds. Our approach is based on the totally geodesic embedding of the target manifold into $\mathbb{R}^N$ . Since embeddings of Riemannian manifolds into Euclidean spaces can easily be made totally geodesic by extending the Riemannian metric in a certain way into some tubular neighbourhood, the here presented approach is quite general. Totally geodesic embeddings allow to reformulate the harmonic map heat flow in a neighbourhood of the embedded target manifold. This reformulation has the advantage that the problem becomes unconstrained: Instead of assuming a priori that the solution to the flow maps into the target manifold this fact becomes a property of the solution to the extended flow for special initial data. The solution space to the reformulated problem therefore exists of maps which are also allowed to map into the ambient space of the target manifold. This simplifies the discretization of the problem. Based on this observation, we here propose algorithms for the computation of the harmonic map heat flow and of harmonic maps. In contrast to previous schemes, our algorithm does not make use of projections onto the target manifold, discrete tangential deformations, geodesic finite elements or of Lagrange multipliers. We prove error estimates in the stationary case and present some numerical tests at the end of the paper.