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 $(T^1{M}^{*},\partial T^1{M}^{*})$ in relation with calibrations and a certain minimal volume equation. A particular family $X_{\text{m},k}$, $k\in \mathbb{N}$, of minimal vector fields on $M^{*}$ is found in an original fashion. The family has unbounded volume, ${\text{lim}}_k\text{vol}(X_{\text{m},k_{|\Omega }})=+\infty$, on any given open subset $\Omega$ of $M^{*}$ and indeed satisfies the necessary differential equation for minimality. Another vector field $X_{\ell }$ is discovered on a region ${\Omega }_1\subset {\mathbb{S}}^2$, with volume smaller than any other known optimal vector field restricted to ${\Omega }_1$.