Vector fields with big and small volume on $\mathbb{S}^2$

We search for minimal volume vector fields on a given Riemann surface, specialising on the case of $M^\star$, this is, the 2-sphere with two antipodal points removed. We discuss the homology theory of the unit sphere tangent bundle $(SM^\star,\partial SM^\star)$ in relation with calibrations and a minimal volume equation. We find a family $X_{\mathrm{m},k},\:k\in\mathbb{N}$, called the meridian type vector fields, defined globally and with unbounded volume on any given open subset $\Omega$ of $M^\star$. In other words, we have that $\forall\Omega$, $\lim_k\mathrm{vol}({X_{\mathrm{m},k}}_{|\Omega})=+\infty$. These are the strong candidates to being the minimal volume vector fields in their homology class, since they satisfy \textit{great} equations. We also show a vector field $X_\ell$ on a specific region $\Omega_1\subset\mathbb{S}^2$ with volume smaller than any other known optimal vector field restricted to $\Omega_1$.