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arXiv · 1910.13966

Harmonic maps from surfaces of arbitrary genus into spheres

Renan Assimos·Jürgen Jost
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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.

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Renan Assimos, Jürgen Jost. 2019-11-04. Harmonic maps from surfaces of arbitrary genus into spheres. https://arxiv.org/abs/1910.13966

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