The geometry of closed conformal vector fields on Riemannian spaces

Abstract

In this paper we examine different aspects of the geometry of closed conformal vector fields on Riemannian manifolds. We begin by getting obstructions to the existence of closed conformal and nonparallel vector fields on complete manifolds with nonpositive Ricci curvature, thus generalizing a theorem of T. K. Pan. Then we explain why it is so difficult to find examples, other than trivial ones, of spaces having at least two closed, conformal and homothetic vector fields. We then focus on isometric immersions, firstly generalizing a theorem of J. Simons on cones with parallel mean curvature to spaces furnished with a closed, Ricci null conformal vector field; then we prove general Bernstein-type theorems for certain complete, not necessarily cmc, hypersurfaces of Riemannian manifolds furnished with closed conformal vector fields. In particular, we obtain a generalization of theorems J. Jellett and A. Barros and P. Sousa for complete cmc radial graphs over finitely punctured geodesic spheres of Riemannian space forms.2000 Mathematics Subject Classification. Primary 53C42, 53C45; Secondary 53C65.

On the geometry of complete spacelike hypersurfaces in the anti-de Sitter space

“…Theorem 3.2). Proceeding, we use an extension of another maximum principle of Yau [31] due to Caminha in [13] to guarantee that a complete spacelike hypersurface of H n+1 1 with bounded mean curvature and constant normalized scalar curvature R < −1 is totally umbilical provided that its Gauss mapping has some suitable behavior (cf. Theorem 3.5).…”

Complete spacelike hypersurfaces with positive r-th mean curvature in a semi-Riemannian warped product

“…In this section, in order to prove our main theorems, we shall make use of the following lemma obtained by A. Caminha [10]. Notice that the following lemma extends a result of S. T. Yau [22] on a version of Stokes theorem for an n-dimensional complete and noncompact Riemannian manifold.…”

“…In particular, L 0 = ∆ and from [10] we know that if M has constant sectional curvature, then L r (f ) = div(P r ∇f ), where div denotes the divergence on Σ n . For a smooth function ϕ : R → R and h ∈ D(Σ n ), it follows from the properties of the Hessian of functions that…”

“…Lemma 5.1 (Proposition 2.1 of [10]). Let X be a smooth vector field on the ndimensional complete, noncompact, oriented Riemannian manifold…”

On complete linear Weingarten hypersurfaces in locally symmetric Riemannian manifolds

“…Proposition 2.1 of [9]; see also the Theorem of Karp [11]). In what follows, divX denotes the divergence of a smooth vector field X ∈ T M .…”