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

Abstract

We search for minimal volume vector fields on a given Riemann surface, specialising on the case of M , this is, the 2-sphere with two antipodal points removed. We discuss the homology theory of the unit sphere tangent bundle (SM , ∂SM ) in relation with calibrations and a minimal volume equation.We find a family X m,k , k ∈ N, called the meridian type vector fields, defined globally and with unbounded volume on any given open subset Ω of M . In other words, we have that ∀Ω, lim k vol(X m,k |Ω ) = +∞. These are the strong candidates to being the minimal volume vector fields in their homology class, since they satisfy great equations. We also show a vector field X on a specific region Ω 1 ⊂ S 2 with volume smaller than any other known optimal vector field restricted to Ω 1 .