Total bending of flows with mean curvature correction

摘要

We prove in this paper that Hopf flows in S~3 are absolute minima of the total bending functional B introduced by G. Wiegmink. They are also absolute minima of the energy functional E introduced by C. M. Wood, once E differs from B by a constant. In fact, we introduce a functional D for a flow on a closed n-dimensional Riemannian manifold M which, in the three dimensional case, coincides with B up to normalization and prove that D is absolutely minimized by Hopf flows on odd-dimensional unit spheres. We also provide an extension of a theorem proven by H. Gluck and W. Ziller about volume of vector fields on S~3.