Abstract
We consider the problem of minimal volume vector fields on a given Riemann surface, specialising on the case of , that is, the arbitrary radius -sphere with two antipodal points removed. We discuss the homology theory of the unit tangent bundle in relation with calibrations and a certain minimal volume equation. A particular family , , of minimal vector fields on is found in an original fashion. The family has unbounded volume, , on any given open subset of and indeed satisfies the necessary differential equation for minimality. Another vector field is discovered on a region , with volume smaller than any other known optimal vector field restricted to .
Funding Statement
The research leading to these results has received funding from Fundação para a Ciência e a Tecnologia. Project Ref. UIDB/04674/2020.
Acknowledgement
We thank Olga Gil-Medrano for helpful conversations and Marta Barata for the drawings. Also, we thank the anonymous Referees’ carefull readings and remarks which led to substantial improvements of this work.
Citation
Rui Albuquerque. "Vector fields with big and small volume on the -sphere." Hiroshima Math. J. 53 (2) 225 - 239, July 2023. https://doi.org/10.32917/h2022009
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