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

2018/04/30 by Loubeau, E., Markellos, M.
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1804.11169

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.

Related

• The biharmonicity of sections of the tangent bundle
• On the biharmonicity of vector fields and unit vector fields
• Biharmonic maps from tori into a 2-sphere
• Biharmonic maps in two dimensions
• A short survey on biharmonic maps between Riemannian manifolds