Harmonic maps from surfaces of arbitrary genus into spheres

TL;DR

This paper explores the existence and construction of harmonic maps from surfaces of any genus into spheres, revealing new examples and geometric properties related to convexity and geodesics.

Contribution

It introduces novel harmonic maps of degree 0 from arbitrary genus surfaces into spheres and identifies regions lacking closed geodesics that still support harmonic maps.

Findings

Constructed new harmonic maps of degree 0 from arbitrary genus surfaces.

Identified regions without closed geodesics that support harmonic maps.

Extended examples building on Kendall's work using Struwe's heat flow method.

Abstract

We relate the existence problem of harmonic maps into $S^2$ to the convex geometry of $S^2$. On one hand, this allows us to construct new examples of harmonic maps of degree 0 from compact surfaces of arbitrary genus into $S^2$. On the other hand, we produce new example of regions that do not contain closed geodesics (that is, harmonic maps from $S^1$) but do contain images of harmonic maps from other domains. These regions can therefore not support a strictly convex function. Our construction builds upon an example of W. Kendall, and uses M. Struwe's heat flow approach for the existence of harmonic maps from surfaces.