Harmonic maps from surfaces of arbitrary genus into spheres

DOI10.1007/S00526-022-02314-4zbMath1503.53128arXiv1910.13966OpenAlexW2982488032WikidataQ115385961 ScholiaQ115385961MaRDI QIDQ2093435

Renan Assimos, Juergen Jost

Publication date: 8 November 2022

Published in: Calculus of Variations and Partial Differential Equations (Search for Journal in Brave)

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.

Full work available at URL: https://arxiv.org/abs/1910.13966

Mathematics Subject Classification ID

Differential geometric aspects of harmonic maps (53C43) Harmonic maps, etc. (58E20)

Cites Work