Volumes of Flows

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 <!-- MATH ${T_ * }\left( M \right)$ --> 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 . 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.