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Volumes of flows

Author: David L. Johnson
Journal: Proc. Amer. Math. Soc. 104 (1988), 923-931
MSC: Primary 53C20; Secondary 58F17
MathSciNet review: 964875
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Abstract: If $ F$ is an oriented nonsingular flow on a Riemannian manifold $ M$, the volume of $ F$ is defined as the $ n$-dimensional measure of the unit vector field tangent to $ F$, as a section of $ {T_ * }\left( M \right)$ 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 $ {S^3}$. 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 $ {S^5}$ exists.

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