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 $(T^1M^\star,\partial T^1M^\star)$ 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 $\Omega_1$.
- Publication:
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arXiv e-prints
- Pub Date:
- October 2021
- DOI:
- 10.48550/arXiv.2110.07759
- arXiv:
- arXiv:2110.07759
- Bibcode:
- 2021arXiv211007759A
- Keywords:
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- Mathematics - Differential Geometry;
- Mathematics - Geometric Topology;
- 53C42;
- 57R25
- E-Print:
- 13 pages