New Bernstein Type Results in Weighted Warped Products

In the last decades, constant mean curvature hypersurfaces in Riemannian manifolds have been deeply studied. ,is is because that such hypersurfaces exhibit nice Bernstein type properties. Indeed, the classical Bernstein theorem states that a complete minimal surface inR3 is a plane. Later, the Bernstein theorem has been extended to higher dimensions as follows: each complete minimal hypersurface in Rn+1 must be hyperplane by Simons in [1] for n≤ 7, and the result was achieved through successive efforts of Almgren [2], Fleming [3] and De Giorgi [4]. However, for n≥ 8, there exists a counterexample in [5], by constructing a nontrivial complete minimal hypersurface in Rn+1, which is not a hyperplane. In recent years, successive efforts have been made in order to extend these Bernstein type results for hypersurfaces to much more general ambient spaces. Among all Riemannian manifolds, we will consider the class of models known as weighted warped products. Our model ambient space will be a warped product I×ρM, in the sense of [6], with an interval I equipped with a positive definite metric as base, a Riemannian manifold M as fiber, and a positive smooth function ρ as warping function. Furthermore, there exists a distinguished family of hypersurfaces in warped products, that is, the slices, which are defined as level hypersurfaces of the height function on the base, as defined in Section 2. Note that each slice is totally umbilical and has constant mean curvature. Moreover, a weighted manifold is a Riemannian manifold with measure that has smooth positive density with respect to the induced metric. More precisely, the weighted manifold Mf associated with a complete Riemannian manifold (M, g) and a smooth positive function f on M is the triple (M, g, dμ � e dM), where dM is the volume element of M. In this setting, we will consider the Bakry–Émery–Ricci tensor (see [7]) which is a generalization of the standard Ricci tensor Ric defined as


Introduction
In the last decades, constant mean curvature hypersurfaces in Riemannian manifolds have been deeply studied. is is because that such hypersurfaces exhibit nice Bernstein type properties. Indeed, the classical Bernstein theorem states that a complete minimal surface in R 3 is a plane. Later, the Bernstein theorem has been extended to higher dimensions as follows: each complete minimal hypersurface in R n+1 must be hyperplane by Simons in [1] for n ≤ 7, and the result was achieved through successive efforts of Almgren [2], Fleming [3] and De Giorgi [4]. However, for n ≥ 8, there exists a counterexample in [5], by constructing a nontrivial complete minimal hypersurface in R n+1 , which is not a hyperplane. In recent years, successive efforts have been made in order to extend these Bernstein type results for hypersurfaces to much more general ambient spaces.
Among all Riemannian manifolds, we will consider the class of models known as weighted warped products. Our model ambient space will be a warped product I× ρ M, in the sense of [6], with an interval I equipped with a positive definite metric as base, a Riemannian manifold M as fiber, and a positive smooth function ρ as warping function. Furthermore, there exists a distinguished family of hypersurfaces in warped products, that is, the slices, which are defined as level hypersurfaces of the height function on the base, as defined in Section 2. Note that each slice is totally umbilical and has constant mean curvature. Moreover, a weighted manifold is a Riemannian manifold with measure that has smooth positive density with respect to the induced metric. More precisely, the weighted manifold M f associated with a complete Riemannian manifold (M, g) and a smooth positive function f on M is the triple (M, g, dμ � e − f dM), where dM is the volume element of M. In this setting, we will consider the Bakry-Émery-Ricci tensor (see [7]) which is a generalization of the standard Ricci tensor Ric defined as So, it is natural to extend some results of the Ricci curvature to similar results for the Bakry-Émery-Ricci tensor. Before presenting more details on our work, we give a brief overview of some results related to our one.
Wei and Wylie researched the weighted Riemannian manifold M f and proved mean curvature and volume comparison results under the assumption that Ric f is bounded from below and f or |∇f| is bounded in [8]. In particular, Salamanca and Salavessa [9] obtained uniqueness results for complete weighted minimal hypersurfaces (that is, those whose weighted mean curvature identically vanishes) in a weighted warped product whose fiber is a parabolic manifold. Later, de Lima et al. [10,11] studied the Bernstein type results concerning complete hypersurfaces in weighted warped products via application of appropriated generalized maximum principles. Furthermore, de Lima et al. [12] obtained Liouville type results for two-sided hypersurfaces in weighted Killing warped products. More recently, the author [13] proved some uniqueness results of complete hypersurfaces in weighted Riemannian warped products I× ρ M n f with f-parabolic fiber, through the application of the weak maximum principle.
Our aim in this paper is to obtain new Bernstein type results for complete constant weighted mean curvature hypersurfaces in weighted warped products. We have organized this article as follows. In Section 2, we introduce some basic notions and facts to be used for hypersurfaces immersed in weighted warped products. In Section 3, we prove some parametric results related to the hypersurfaces in a weighted warped product that will enable us to obtain our main uniqueness result ( eorem 1) which extend the corresponding results in [13]. To conclude this paper, we will devote Section 4 to prove a new Bernstein type result for entire graphs in weighted warped products for the constant weighted mean curvature case ( eorem 2).

Preliminaries
Let M n be a connected n( ≥ 2)-dimensional oriented Riemannian manifold and I ⊂ R be an open interval endowed with the metric dt 2 . Let ρ: I ⟶ R + be a smooth function. Denote by I× ρ M n the warped product manifold with the Riemannian metric where π I and π M are the projections onto I and M, respectively. is resulting space is a warped product in the sense of [6], with base (I, dt 2 ), fiber (M, 〈, 〉 M ), and warping function ρ. Furthermore, for every t 0 ∈ I, we say that Consider the vector field K � ρ(π I )z t in I× ρ M n , where z t � (z/zt) is the unit vector field tangent to base I. Moreover, using the relationship between the Levi-Civita connections of I× ρ M n and those of the fiber and the base (see Corollary 7.35 in [6]), we have for any X ∈ X(I× ρ M n ), where ∇ stands for the Levi-Civita connection of the Riemannian metric 〈, 〉 in (2). erefore, K is conformal with L K 〈, 〉 � 2ρ ′ (π I )〈, 〉 and its metrically equivalent 1-form is closed.
Recall that a smooth immersion ψ: Σ n ⟶ I× ρ M n of an n-dimensional connected manifold Σ n is said to be a hypersurface. Moreover, the induced metric via ψ on Σ n will be also denoted by 〈, 〉.
In this paper, we study the connected hypersurfaces ψ: Σ n ⟶ I× ρ M n oriented a unit normal vector field N. Let ∇ be the Levi-Civita connection of Σ n . e Gauss and Weingarten formulas for the hypersurfaces ψ: Σ n ⟶ I× ρ M n are given, respectively, by where X, Y ∈ X(Σ n ) and A: X(Σ n ) ⟶ X(Σ n ) is the shape operator (or Weingarten endomorphism) of Σ n with respect to N.
In the following, we consider two particular functions naturally attached to hypersurface Σ n , namely, the angle (or support) function Θ � 〈N, z t 〉 and the height function h � (π I )| Σ n .
Let ∇ and ∇ be the gradients with respect to the metrics of I× ρ M n and Σ n , respectively. en, by a simple computation, we obtain So, the gradient of h on Σ n is Particularly, where | · | denotes the norm of a vector field on Σ n . Moreover, taking tangential components in (3), we have, from (4) and (5), that where N〉N is the tangential component of K along Σ n . is enables us to use (9) to compute the gradient of the angle function Θ, obtaining Furthermore, it follows from (7) and (9) that the Laplacian of h on Σ n is Consequently, by Δρ(h) � ρ ′ (h)Δh + ρ ″ (h)|∇h| 2 , we have Furthermore, consider that a warped product I× ρ M n endowed with a weight function f, which will be called a weighted warped product I× ρ M n f . In this setting, for a hypersurface Σ n immersed into I× ρ M n f , the f-divergence operator on Σ n is defined by where X is a tangent vector field on Σ n . For a smooth function u: Σ n ⟶ R, we define its drifting Laplacian by and we will also denote such an operator as the f-Laplacian of Σ n .
According to Gromov [14], the weighted mean curvature or f-mean curvature H f of Σ n is given by where H is the standard mean curvature of hypersurface Σ n with respect to N.
In this paper, we will consider weighted Riemannian warped products I× ρ M n f whose weight function f does not depend on the parameter t ∈ I, that is, 〈∇ f, z t 〉 � 0. Moreover, we will refer to them as M n+1 : � I× ρ M n f .

Parametric Uniqueness Results
In order to prove our uniqueness results in weighted warped product M n+1 , we need a few previous results.
where Ric M f stands for the Bakry-Émery-Ricci curvature tensor of M n and N * � N − 〈N, z t 〉z t is the projection of the vector field N onto M n . Proof.
e key idea of the proof is to compute the f-Laplacian of the function Θ. To do so, taking into account (14), it follows that Proceeding as above in Section 2, by a direct computation from (10) and (12), gives Moreover, from a straightforward computation, we obtain So, by (14) and (15), we can write (17) as On the contrary, Lemma 1 in [11] proves that Journal of Mathematics where Ric M stands for the Ricci curvature tensor of M n . Moreover, taking into account that 〈∇ f, z t 〉 � 0, it is easy to obtain that Hence, using relation (1), the result follows from (20)-(22) that Moreover, from (10), we obtain that To conclude the proof, we should notice that and use (23) and (24) to obtain (16).
In the following, we give the next technical lemma (for further details, see Proposition 3.3 in [13]) which will be essential for the proofs of our main results. Moreover, we also need to consider the weighted warped products I× ρ M n f satisfy the following convergence condition: Moreover, using Young's inequality, we have So, we can estimate Considering the assumptions of eorem 1, it follows from (16) that Moreover, |∇h| 2 is bounded on Σ n which allows us to apply Lemma 2 to guarantee that |∇h| 2 is constant. So, Δ f |∇h| 2 � 0. From (16), we also have that f-mean curvature such that H f ρ ′ (h)Θ ≥ 0. Suppose that either inequality (26) is strict or ρ ″ (h) > 0, then, for constant /3), the only bounded entire solutions to equation (39) with |Du| < λρ(u) are the constant ones u � t 0 for some t 0 ∈ I.
(41) e rest of our assumptions enable us to apply eorem 1 to end the proof.

Data Availability
No data were used to support this study.

Conflicts of Interest
e author declares that there are no conflicts of interest regarding the publication of this paper.