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volume functional — 1 results found.

      
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On the volume of unit vector fields on Riemannian three-manifolds
C. R. Math. Rep. Acad. Sci. Canada Vol. 30 (1) 2008, pp. 11–21
Domenico Perrone (Received: 2007/03/08, Revised: 2008/01/22)

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H. Gluck and W. Ziller proved that the Hopf vector fields, namely, the unit Killing vector fields, are the unique unit vector fields on the unit sphere \(S^3\) that minimize the functional volume. The authors proved this important and famous result by using the method of “calibrated geometries” of Federer and Harvey–Lawson. In this paper, by using a different method, we get an analogue of Gluck and Ziller’s theorem for a compact Sasakian three-manifold with Webster scalar curvature \(w\geq 1\). Moreover, our method gives a new proof of Gluck and Ziller’s theorem. We also extend a theorem of F. Brito about the energy of unit vector fields.

H. Gluck et W. Ziller prouvèrent que les champs de Hopf, c’est-á-dire, les champs vectoriels unitaires de Killing, sont les seuls champs vectoriels unitaires sur la sphére unitaire \(S^3\) que minimisent le volume fonctionnel. Ils prouvèrent cet résultat important en utilisant la méthode des “géométries calibrées” de Federer et Harvey–Lawson. Dans cet article, en utilisant une méthode différente, nous obtenons l’analogue du théorème de Gluck et Ziller pour une 3-variété compact de Sasaki avec courbure scalaire de Webster \(w\geq 1\). En outre, notre méthode donne une nouvelle démonstration du théorème de Gluck et Ziller. Nous aussi étendons un théorème de Brito concernant l’énergie de champs vectoriels unitaires.

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