The geometry of closed conformal vector fields on Riemannian spaces (Q420547)

The author examines 3 different aspects of conformal vector fields on a Riemannian manifold: obstructions to their existence, their use to generate isometric immersions with parallel mean curvature, and their role in Bernstein type results. In Section 2, a result of \textit{S.-T. Yau} [Indiana Univ. Math. J. 25, 659--670 (1976); Erratum ibid. 31, 607 (1982; Zbl 0335.53041)] is refined and applied to the study of closed conformal vector fields on complete non compact Riemannian manifolds with non positive Ricci curvature. Assuming the vector field satisfies a suitable integrability condition, it is shown to be both parallel and a direction of vanishing of the Ricci curvature of the manifold. This generalizes a theorem of \textit{T. K. Pan} [Proc. Am. Math. Soc. 14, 653--657 (1963; Zbl 0131.19603)]. In Section 3, a general geometric construction of closed conformal vector fields on certain hypersurfaces of ambient spaces is given. In Section 4, a theorem of \textit{J. Simons} [Ann. Math. (2) 88, 62--105 (1968; Zbl 0181.49702)] on cones with parallel mean curvature on Euclidean spaces furnished with a closed conformal vector field is given. Section 5 deals with Bernstein type results where a result of Jellett as extended by \textit{A. Barros} and \textit{P. Sousa} [Bull. Sci. Math. 133, No. 2, 190--197 (2009; Zbl 1162.53044)] for complete CMC radial graphs over a finitely punctured sphere in Euclidean space is generalized.