Biharmonic maps from tori into a 2-sphere

TL;DR

This paper investigates biharmonic maps from tori into spheres, classifying their existence under various metrics and showing that no proper biharmonic maps of degree ±1 exist in many cases, extending known harmonic map results.

Contribution

It provides new classifications of biharmonic maps from tori into spheres with different metrics, highlighting the non-existence of certain proper biharmonic maps in these settings.

Findings

No proper biharmonic maps of degree ±1 in many cases

Classified biharmonic maps from tori with flat or non-flat metrics

Extended non-existence results from harmonic to biharmonic maps

Abstract

Biharmonic maps are generalizations of harmonic maps. A well-known result of Eells and Wood on harmonic maps between surfaces shows that there exists no harmonic map from a torus into a sphere (whatever the metrics chosen) in the homotopy class of maps of Brower degree $\pm 1$. It would be interesting to know if there exists any biharmonic map in that homotopy class of maps. In this paper, we obtain some classifications on biharmonic maps from a torus into a sphere, where the torus is provided with a flat or a class of non-flat metrics whilst the sphere is provided with the standard metric. Our results show that there exists no proper biharmonic maps of degree $\pm 1$ in a large family of maps from a torus into a sphere.