Volumes of flows

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.