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Uniqueness of entire graphs in Riemannian warped products

In this paper, by applying the generalized Omori–Yau maximum principle for complete spacelike hypersurfaces in warped product spaces, we obtain the sign relationship between the derivative of warping function and support function. | Turk J Math (2016) 40: 1246 – 1257 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ doi:10.3906/mat-1510-2 Research Article Uniqueness of entire graphs in Riemannian warped products Junhong DONG, Ximin LIU∗ School of Mathematical Sciences, Dalian University of Technology, Dalian, P.R. China Received: 01.10.2015 • Accepted/Published Online: 03.02.2016 • Final Version: 02.12.2016 Abstract: In this paper, by applying the generalized Omori–Yau maximum principle for complete spacelike hypersurfaces in warped product spaces, we obtain the sign relationship between the derivative of warping function and support function. Afterwards, by using this result and imposing suitable restrictions on the higher order mean curvatures, we establish uniqueness results for the entire graph in a Riemannian warped product space, which has a strictly monotone warping function. Furthermore, applications to such a space are given. Key words: Warped product, monotonic function, complete spacelike hypersurface, r -th mean curvature 1. Introduction In recent years, there has been steadily growing interest in the study of hypersurfaces immersed into a Riemannian product space R ×f M n . A basic question on this topic is the problem of uniqueness of spacelike hypersurfaces with some suitable restriction on the mean curvature, more generally, on the higher order mean curvatures. Before giving details of our work we present a brief outline of some recent results related to it. Some works have studied hypersurfaces with constant mean curvature (more generally, constant higher order mean curvatures) immersed in warped product spaces. In [15] Montiel studied the uniqueness of constant mean curvature compact hypersurfaces immersed in warped products of the type R ×f M n and S1 ×f M n , whose Ricci curvature RicM of the fiber M n and the warping function f satisfy the following convergence condition RicM ≥ (n − 1) supR (f ′2 − f ′′ f )⟨ , ⟩M . Later, in [2,