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 $M^{\star }$, that is, the arbitrary radius 2-sphere with two antipodal points removed. We discuss the homology theory of the unit tangent bundle $(T^1{M}^{\star },\partial T^1{M}^{\star })$ in relation with calibrations and a certain minimal volume equation. A particular family $X_{\mathrm{m},k},k\in \mathbb{N}$, of minimal vector fields on $M^{\star }$ is found in an original fashion. The family has unbounded volume, ${\lim }_k\mathrm{vol}({X_{\mathrm{m},k}}_{|\Omega })=+\infty$, on any given open subset $\Omega$ of $M^{\star }$ 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 \textit{optimal} vector field restricted to ${\Omega }_1$.

13 pages; final version, accepted for publication in Hiroshima Mathematical Journal

Cited by in corpus (1)

• The volume of a unit vector field in 2 dimensions via calibrations