The biharmonic homotopy problem for unit vector fields on 2-tori

TL;DR

This paper investigates biharmonic unit vector fields and sections on 2-tori, providing characterizations under conformal changes, proving biharmonic sections are harmonic, and establishing existence results for vector fields in each homotopy class.

Contribution

It offers new characterizations of biharmonic unit vector fields and sections on surfaces, especially on 2-tori, and proves existence theorems for biharmonic unit vector fields in each homotopy class.

Findings

Biharmonic unit sections on 2-tori are always harmonic.

General existence theorem for biharmonic unit vector fields in each homotopy class.

Characterizations of biharmonic unit vector fields under conformal changes.

Abstract

The bienergy of smooth maps between Riemannian manifolds, when restricted to unit vector fields, yields two different variational problems depending on whether one takes the full functional or just the vertical contribution. Their critical points, called biharmonic unit vector fields and biharmonic unit sections, form different sets. Working with surfaces, we first obtain general characterizations of biharmonic unit vector fields and biharmonic unit sections under conformal change of the metric. In the case of a 2-dimensional torus, this leads to a proof that biharmonic unit sections are always harmonic and a general existence theorem, in each homotopy class, for biharmonic unit vector fields.