On Bernstein ’ s Problem of Complete Parabolic Hypersurfaces in Warped Products

We study constant mean curvature hypersurfaces constructed over the ﬁ ber M n of warped products I × f M n . In this setting, assuming that the sign of the angle function does not changed along the hypersurfaces, we infer the uniqueness of such hypersurfaces by applying a parabolicity criterion. As an application, we get some Bernstein type theorems.


Introduction
In this paper, we investigate the uniqueness results in a certain class of Riemannian manifolds, that is, the warped products. In the sense of [1], the warped products I × f M n , where the base I ⊂ ℝ is an interval, the n-dimensional Riemannian manifold M n is a fiber, and f : I ⟶ ℝ + is a warping function (for further details, see Section 2). Before presenting details on our work, we give a brief overview of some articles concerning ours.
In [2], Montiel proved that any compact orientable constant mean curvature hypersurfaces in warped products ℝ × ρ M n that can be written as a graph over M n must be a slice, under the assumption that the Ricci curvature Ric M of M n and the function ρ satisfy the convergence condition Ric M ≥ ðn − 1Þ sup ðρ′ 2 − ρρ′′Þ. Later on, Alas and Dajczer [3] studied the constant mean curvature hypersurfaces in warped product spaces. In this setting, if the hypersurface is compact, they extended the previous results by Montiel. Afterwards, some of these generalizations hold for complete hypersurfaces. In recent years, by using the Omori-Yau generalized maximum principle for complete hypersurfaces and supposing suitable assumptions, some researchers proved that such a hypersurface must be a slice. For instance, in [4], Caminha and de Lima studied complete graphs of constant mean curvature in the hyperbolic and steady-state spaces, and they obtained some rigidity theorems for such graphs. Later, Aquino and de Lima [5] extended the results in [4] to complete constant mean curvature graphs in warped products under appropriate convergence condition. In [6], Cavalcante et al. considered the Bernstein type properties of complete two-sided hypersurfaces in weighted warped products; they established sufficient conditions which guarantee that such a hypersurface must be a slice. Furthermore, [7] obtained uniqueness results for complete hypersurfaces in Riemannian warped products whose fiber has parabolic universal covering. More recently, by the weak Omori-Yau's maximum principle, the author [8] proved new Bernstein type results of complete constant weighted mean curvature hypersurfaces in weighted warped products I × ρ M n f . This paper is organized as follows: in Section 2, we introduce some basic facts for hypersurfaces in warped products. Section 3 is devoted to compute the Laplacian of the angle function Θ which we will define in Section 2. Moreover, using the parabolicity criterion, we establish the uniqueness results concerning constant mean curvature hypersurfaces. As a consequence of this previous study, we prove some Bernstein type results for constant mean curvature entire graphs in warped products.

Preliminaries
Throughout this paper, we consider the warped products M n+1 = I × f M n , where M n is a connected oriented n -dimensional Riemannian manifold, I ⊂ ℝ is an interval with a positive definite metric dt 2 , f : I ⟶ ℝ + is a positive smooth function, and the product manifold I × M n is endowed with the Riemannian metric where π I and π M denote the projections onto I and M, respectively. Such the resulting space is said to be a warped product in [1], Chapter 7, with fiber ðM n , h,i M Þ, base ðI, dt 2 Þ, and warping function f . Moreover, an immersion ψ : Σ n ⟶ M n+1 of an n-dimensional manifold Σ n is called a hypersurface. Furthermore, the induced metric through ψ on Σ n is also denoted by h,i.
In fact that each leaf M n t 0 = ft 0 g × M n of the foliation t 0 ∈ I ⟶ M n t 0 of M n+1 by complete hypersurfaces has constant where ∂ t = ∂/∂t is a unit vector field tangent to I, X ∈ X ð M n+1 Þ, and ∇ is the Levi-Civita connection in M n+1 .
In this paper, we consider the hypersurfaces ψ : Σ n ⟶ M n+1 oriented a unit normal vector field N, and such hypersurfaces are called two-sided hypersurfaces. In what follows, we will study its angle (or support) function Θ = hN, ∂ t i and height function τ = ðπ I Þj Σ n .
Let ∇ be the Levi-Civita connection in Σ n . A direct computation shows that Thus, the gradient of τ is given by where ð·Þ Τ is the tangential component of a vector field in Xð M n+1 Þ along Σ n . Moreover, where j·j is the norm of a vector field on Σ n .

Parametric Uniqueness Results
In order to establish our uniqueness results in warped products M n+1 , we need to compute the Laplacian of the angle function Θ to obtain a bounded on the Laplacian of the function log ð1 + ΘÞ.
where Ric M is the Ricci curvature tensor of M n and N * = N − hN, ∂ t i∂ t stands for the projection of the vector field N onto M n .
Proof. The Gauss and Weingarten formulas of ψ : where X, Y ∈ XðΣÞ and A : XðΣÞ ⟶ XðΣÞ is the shape operator of Σ n corresponding to N. Now, taking the tangential component in (2), by (7) and (8), we can easily get that where f ′ðτÞ = f ′ ∘ τ and K Τ = f ðτÞ∂ Τ t = K − hK, NiN is the tangential component of K along Σ n . Therefore, it follows from (9) that Moreover, by (4) and (9), we conclude that Let E 1 , ⋯, E n be a local orthonormal frame on XðΣ n Þ; from (10), we know that In fact, for every X, Y ∈ XðΣ n Þ, we have that

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where ∇ X A denotes the covariant derivative of A. From (12) and (13), we can rewrite (12) as Recall that the Codazzi equation of ψ : or, equivalently, where R denotes the curvature tensor of M n+1 . Therefore, using (11) and (15), we conclude from (14) that Here, we know the general fact that for every X ∈ XðΣ n Þ. Then, it follows from (9) that where Ric stands for the Ricci curvature tensor of M n+1 . On the other hand, using Corollary 7.43 of [1], we have that where Ric M is the Ricci curvature tensor of the fiber M n and N * = N − hN, ∂ t i∂ t and ð∇τÞ * = ∇τ − h∇τ, ∂ t i∂ t denote the projections of the vector field N and ∇τ onto M n , respectively. Moreover, from (4), we obtain Therefore, from (20) and (21), we get Finally, substituting (22) into (19), by a direct computation, we can obtain (6).
We recall that a complete Riemannian manifold is parabolic in the sense that any positive superharmonic function on the Riemannian manifold must be constant (see [9]). In this setting, we obtain some uniqueness results via parabolicity criterion. Proof. From (10), we obtain that Therefore, using (6) and (23), we deduce that

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Moreover, using Young's inequality, we have So, we can estimate Using the hypothesis of Theorem 2, we conclude that log ð1 + ΘÞ > 0 and Δ log ð1 + ΘÞ ≤ 0, which suffices to show that log ð1 + ΘÞ = log ð1 + Θ 0 Þ is constant, So, Δ log ð1 + ΘÞ = 0. From (26), we know that It follows that the hypersurface Σ n is totally geodesic and f ðτÞ is constant. In addition, if the fiber M n has positive Ricci curvature, from (26), we have that N * ðpÞ = 0 at any p ∈ Σ n ; that is, ∇τ = 0 on Σ n , and then, Σ n is a totally geodesic slice. Moreover, we note that the mean curvature H of a slice in a warped product I × f M n is given by Therefore, Σ n is a totally geodesic minimal slice.