Hypersurfaces of constant higher order mean curvature in warped products

Abstract

In this paper we characterize compact and complete hypersurfaces with some constant higher order mean curvature into warped product spaces. Our approach is based on the use of a new trace operator version of the Omori-Yau maximum principle which seems to be interesting in its own.

Uniqueness of entire graphs in warped products

“…where Ric M is the Ricci tensor of the fiber M n (see, for instance, [3,6,17]). Warped product spaces are foliated by slices, M t = {t} × M n , which are hypersurfaces with constant higher order mean curvature.…”

“…Some of these generalizations hold even for complete, non-compact hypersurfaces. More recently, by using a suitable generalized version of the Omori-Yau maximum principle for an appropriate tracetype differential operator, the first author jointly with Impera and Rigoli considered in [6] the uniqueness problem for immersed hypersurfaces with some constant higher order mean curvature in warped product spaces.…”

“…According to Lemma 25 of [6] (see also Lemma 3.1 of [5], paying attention to the opposite convention for the sign of R in [5]), the divergence of the Newton transformations are given by…”

On complete linear Weingarten hypersurfaces in locally symmetric Riemannian manifolds

“…Consequently, we have that div(P 1 (∇f )) = On the other hand, since M n+1 is an Einstein manifold with n ≥ 3, there exists a parameter λ ∈ R such that Ric = λ , , where Ric denotes the Ricci tensor of M n+1 . Thus, denoting by R the curvature tensor of M n+1 , from Lemma 25 of [5] (see also Lemma 3.1 of [2]) we have divP 1 , ∇f = i R(η, e i )∇f, e i = Ric(η, ∇f ) = λ η, ∇f = 0, where η stands for the unit normal vector field on M n . Hence, from (4.13), we conclude that (4.14) f = div(P 1 (∇f )).…”

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

“…They also investigated the uniqueness of complete spacelike hypersurfaces by using a generalization of the Omori-Yau maximal principal. We also refer the reader to [2,4] for some relevant results concerning higher order mean curvature.…”