Energy and volume of vector fields on spherical domains (Q442038)

It has been known by work of \textit{H. Gluck} and \textit{W. Ziller} [Comment.\ Math.\ Helv.\ 61, 177--192 (1986; Zbl 0605.53022)] and \textit{F. G. B. Brito} [Differ.\ Geom.\ Appl. 12, 157--163 (2000; Zbl 0995.53023)] that unit vector fields on \(S^3\) of minimum volume or energy are precisely the Hopf vector fields. This is generalized here to spherical domains on odd-dimensional spheres, as follows.NEWLINENEWLINELet \(K\) be a connected open subset in \(S^{2k+1}\) and \(v\) a unit vector field on \(K\) which coincides with a Hopf flow on the (sufficiently smooth) boundary. Then the energy of \(v\) is bounded from below by \(({2k+1\over2}+{k\over 2k-1})\) times the volume of \(K\), and for the volume of \(v\) a similar estimate holds with constant \(4^k/{2k\choose k}\).