Volumes of flows

Author:
David L. Johnson

Journal:
Proc. Amer. Math. Soc. **104** (1988), 923-931

MSC:
Primary 53C20; Secondary 58F17

DOI:
https://doi.org/10.1090/S0002-9939-1988-0964875-4

MathSciNet review:
964875

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Abstract: If is an oriented nonsingular flow on a Riemannian manifold , the *volume* of is defined as the -dimensional measure of the unit vector field tangent to , as a section of with the induced metric. It is shown that, for any metric of the two-dimensional torus, and for any homotopy class of flows on the torus, there is a unique smooth flow of minimal volume within the homotopy class. It has been shown that the Hopf foliation on the round threesphere absolutely minimizes the volume of flows on . In higher dimensions this is not the case; the Hopf fibrations are not even local minima of the volume functional for flows on the round five-sphere. It is not known whether a volume-minimizing flow on exists.

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DOI:
https://doi.org/10.1090/S0002-9939-1988-0964875-4

Article copyright:
© Copyright 1988
American Mathematical Society