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

TL;DR

This paper investigates minimal volume vector fields on a punctured 2-sphere, introducing a novel family with unbounded volume and identifying a vector field with minimal volume on a specific region.

Contribution

It presents a new family of minimal vector fields on the punctured 2-sphere and analyzes their volume properties, including unboundedness and optimality in certain regions.

Findings

Family $X_{m,k}$ has unbounded volume on any open subset.

A vector field $X_ ext{l}$ with minimal volume on a region $oldsymbol{ ext{Ω}_1}$ is identified.

The work relates homology theory of the tangent bundle with minimal volume equations.

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 $(T^1M^\star,\partial T^1M^\star)$ 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 $\Omega_1$.