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Total bending of flows with mean curvature correction. (English) Zbl 0995.53023
The author introduces the functional \[ D(\vec{v}) = \int_M \left( \|\nabla \vec{v} \|^2 +\;(n-1) (n-3) \|\vec{H}_{\vec{v}_{\bot}}\|^2 \right) dM, \] on the unit vector fields \(\vec v\) of a closed Riemannian manifold \(M\), where \(\vec{H}_{\vec{v}_{\bot}}\) is the mean curvature vector of the distribution \(\vec{v}_{\bot}\), orthogonal to \(\vec{v}\). He shows that Hopf flows uniquely minimize \(D\) on odd-dimensional unit spheres. On \(S^3\) the functional \(D\) coincides, up to a constant, with the total bending of a unit vector field as defined by G. Wiegmink. Therefore it follows that Hopf flows minimize \(B\) on \(S^3\) and, moreover, these are the unique absolute minima of \(B\).
The paper also extends a result of H. Gluck and W. Ziller on the lower bound for the volume of vector fields on \(S^3.\)

MSC:
53C20 Global Riemannian geometry, including pinching
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