Vector fields with big and small volume on the 2-sphere

Abstract

We consider the problem of minimal volume vector fields on a given Riemann surface, specialising on the case of M*, that is, the arbitrary radius 2-sphere with two antipodal points removed. We discuss the homology theory of the unit tangent bundle (T^1M*,∂T^1M*) in relation with calibrations and a certain minimal volume equation. A particular family X_m,k , k ∈ N, of minimal vector fields on M* is found in an original fashion. The family has unbounded volume, lim_k vol(X_m,k|Ω)=+∞, on any given open subset Ω of M* and indeed satisfies the necessary differential equation for minimality. Another vector field X_l is discovered on a region Ω_1 ⊂ S^2, with volume smaller than any other known optimal vector field restricted to Ω_1.