Uniqueness of Complete Hypersurfaces in Weighted Riemannian Warped Products

In this paper, applying the weak maximum principle, we obtain the uniqueness results for the hypersurfaces under suitable geometric restrictions on the weighted mean curvature immersed in a weighted Riemannian warped product I × ρMf whose fiber M has f -parabolic universal covering. Furthermore, applications to the weighted hyperbolic space are given. In particular, we also study the special case when the ambient space is weighted product space and provide some results by Bochner’s formula. As a consequence of this parametric study, we also establish Bernstein-type properties of the entire graphs in weighted Riemannian warped products.


Introduction
In recent years, the study of complete hypersurfaces in Riemannian manifolds has attracted many geometers. This is due to the fact that such hypersurfaces exhibit nice Bernstein-type properties.
Particularly, from the geometric analysis point of view, many problems lead us to consider Riemannian manifolds with a measure that has a positive smooth density with respect to the Riemannian one. This turns out to be compatible with the metric structure of the manifold, and the resulting spaces are the weighted manifolds, which are also called manifolds with density or smooth metric measure spaces. More precisely, the weighted manifold M n f is associated with a complete n-dimensional Riemannian manifold ðM n , gÞ, and a smooth function f on M n is the triple ðM n , g, dμ = e −f dMÞ, where dM stands for the volume element of M n . In this setting, we will take into account the so-called Bakry-Émery Ricci tensor (see [1]) which is an extension of the standard Ricci tensor Ric, which is defined by So, it is natural to extend some results of the Ricci curvature to analogous results for the Bakry-Émery Ricci tensor.
Before giving more details on our work, we present a brief outline of some recent results related to ours.
In [2], Wei and Wylie studied the complete n-dimensional weighted Riemannian manifold and proved the weighted mean curvature and volume comparison results under the ∞-Bakry-Émery Ricci tensor is bounded from below and f or |∇f | is bounded. Later, de Lima et al. [3,4] researched the uniqueness of complete two-sided hypersurfaces immersed in weighted warped products by applying the appropriated generalized maximum principles. Moreover, [5] established Liouville-type results related to twosided hypersurfaces immersed in a weighted Killing warped product. More recently, some uniqueness results of complete two-sided hypersurfaces in warped products with density are given in [6].
In this paper, we study complete hypersurfaces in a weighted Riemannian warped product. The Riemannian warped product I × ρ M n where I ⊂ ℝ is an open interval, M n is a complete n-dimensional Riemannian manifold, ρ : I ⟶ ℝ + is a positive smooth warping function, and the warped metric is given by where h,i M is the metric tensor of M n . Furthermore, there exists a distinguished family of hypersurfaces in Riemannian warped products, that is so-called slices, which are defined as level hypersurfaces of the coordinate of the space. Notice that any slice is totally umbilical and has constant mean curvature. This manuscript is organized as follows. In Section 2, we introduce some basic notions and facts of the hypersurfaces immersed in weighted Riemannian warped products. Section 3 is devoted to prove some results concerning the f -parabolicity of weighted manifolds and pay attention to show the weak maximum principle for the f -Laplace operator Δ f holds on f -parabolic weighted manifolds. Moreover, by using the weak maximum principle, we provide the sign relationship among the f -mean curvature and the derivative of the warping function. These auxiliary results will be the key to obtaining our results. In our main results, we establish the uniqueness results for complete hypersurfaces under appropriate conditions on the f -mean curvature and the warping function in weighted Riemannian warped products M n+1 = I × ρ M n f whose fiber M n f has f -parabolic universal covering. Besides, we also present some applications related to our results. In Section 4, applying the weak maximum principle and Bochner's formula, we obtain some rigidity results for the special case when the ambient space is weighted product space. Section 5, as a nondirect application of our parametric case, we get nonparametric results for the entire graphs in weighted Riemannian warped products.

Preliminaries
Let M n be a connected n-dimensional oriented Riemannian manifold and I ⊂ ℝ be an open interval which is endowed with the metric dt 2 . Let ρ : I ⟶ ℝ + be a positive smooth function. Denote I × ρ M n to be the product manifold with the following Riemannian metric where π I and π M are the projections onto I and M, respectively. Following the terminology used in [7], Chap.7, this resulting space is a warped product with fiber ðM, h,i M Þ, base ðI, dt 2 Þ, and warping function ρ. Furthermore, for a fixed point t 0 ∈ I, we say that M n t 0 = ft 0 g × M n is a slice of I × ρ M n . Recalling that a smooth immersion ψ : Σ n ⟶ I × ρ M n of an n-dimensional connected manifold Σ n is called to be a hypersurface. Moreover, the induced metric via ψ on Σ n will be also denoted for h,i.
Throughout this paper, we assume that Σ n is a two-sided hypersurface. Recalling that a hypersurface Σ n is called a twosided hypersurface if its normal bundle is trivial, which means that there exists a globally defined unit normal vector field N ∈ X ⊥ ðΣÞ. For instance, every hypersurface with never vanishing mean curvature is trivially two-sided. Moreover, when the hypersurface Σ n is two-sided, a choice of N on Σ n makes the second fundamental form globally defined on X ⊥ ðΣÞ. In the sequel, the Riemannian warped product is clearly orientable. This allows us to take, for each two-sided hypersurface Σ n , a unique unitary normal vector field N globally defined on Σ n in the same orientation of the vector field ∂ t , ∂ t ≔ ∂/∂t, i.e., such that hN, ∂ t i ≤ 0. By the wrongway Cauchy-Schwarz inequality (see [7], Proposition 5.30), we have −1 ≤ hN, ∂ t i ≤ 0, and the first equality holds at a point p ∈ Σ n if and only if N = −∂ t at p. Moreover, we will refer to the function Θ : Σ ⟶ ½−1, 0, Θ ≔ hN, ∂ t i, as the angle function. On the other hand, we will represent a particular function naturally attached to the hypersurface Σ n by the height function h = ðπ I Þj Σ : Σ n ⟶ I.
It can be easily seen that a hypersurface in Riemannian warped products is a slice if and only if the height function is constant. We also observe that slice ft 0 g × M n of I × ρ M n has constant mean curvature H = ρ ′ ðt 0 Þ/ρðt 0 Þ with respect to the unit normal vector field N = −∂ t .
Let ∇ and ∇ stand for gradients with respect to the metrics of I × ρ M n and Σ n , respectively. In a simple computation, we have So, the gradient of h on Σ n is Particularly, we have where || denotes the norm of a vector field on Σ n . Now, we consider that a Riemannian warped product I × ρ M n endowed with a weighted function f , which will be called a weighted Riemannian warped product I × ρ M n f . In this setting, for a two-sided 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 we will also denote such an operator as the f -Laplacian of Σ n .
According to Gromov [8], 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 the Gauss map N.
Notice it follows from a splitting theorem by the case (see [9], Theorem 1.2) that if a weighted Riemannian warped product I × ρ M n f is endowed with a bounded weighted 2 Advances in Mathematical Physics function f and such that Ric f ðV, VÞ ≥ 0 for all vector fields V on I × ρ M n f , then f must be constant along ℝ. So, motivated by this result, in the following, we will consider weighted Riemannian warped products I × ρ M n f whose weighted function f does not depend on the parameter t ∈ I, that is h ∇f , ∂ t i = 0. Moreover, for simplicity, we will refer to them as

Remark 1.
We note that the f -mean curvature H f of a slice ft 0 g × M n in a weighted Riemannian warped product M n+1 is given by, Indeed, since −∂ t = N is a normal vector field to the slice ft 0 g × M n , from (9), we have that H f = H = ρ′ðt 0 Þ/ρðt 0 Þ.
For the proof of our main results in this paper, we need the following formulas that will be the extensions of Lemma 2 in [10].

Lemma 2 ([10]
). Let ψ : Σ n ⟶ I × ρ M n f be a hypersurface immersed in a weighted Riemannian warped product, with height function h. Then where σðtÞ is a primitive function of ρðtÞ.
In the following terminology introduced in [11], we present the definition of the weak maximum principle for the drifted Laplacian. The next lemma extended the result of [11].
Equivalently, for any smooth bounded below function u on M, then there is a sequence fq j g ⊂ M such that On the other hand, a smooth function u on a weighted manifold M f is called f -superharmonic if Δ f u ≤ 0. Taking this into account, a noncompact weighted manifold ðM n , ; ; e −f dMÞ is said to be f -parabolic if it does not admit nonconstant positive f -superharmonic functions on M. So, we can conclude the following extension of Theorem 1 in [12], which establishes sufficient conditions to ensure that the two-sided hypersurface Σ n in M n+1 is f -parabolic.

Lemma 4 ([12]
). Let Σ n be a complete two-sided hypersurface in a weighted Riemannian warped product I × ρ M n f whose fiber M has f -parabolic universal covering. If the angle function Θ is bounded and the restriction ρðhÞ on Σ n of warping function ρ satisfies: (c 1 ) sup ρðhÞ < ∞ (c 2 ) inf ρðhÞ > 0 then, Σ n is f -parabolic.

Uniqueness Results in Weighted Riemannian Warped Products
In this section, we will study the uniqueness for complete hypersurfaces in weighted Riemannian warped products M n+1 . Before describing our main results, we will prove some auxiliary propositions which will be essential in the sequel. Proof. Since weighted manifold M is f -parabolic, using Corollary 6.4 in [13] it follows that M is also stochastically complete.
On the other hand, by the fact which in [11] that M satisfies the weak maximum principle for the f -Laplace operator Δ f if and only if M is stochastically complete, this concludes the proof.
Furthermore, for any compact subset Ω ⊂ ðM n , h,i M , e −f dMÞ, we define the f -capacity of Ω as, where Lip f ðMÞ is the set of all compactly supported Lipschitz functions on M. By the fact that a weighted manifold is f -parabolic if and only if cap f ðΩÞ = 0 for any compact set Ω.
The following lemma is the extension of Lemma 3 in [14], which will allow us to obtain our technical result. For any r such that 0 < r < R, we have where B r denotes the geodesic ball of radius r around p ∈ M and 1/μ r,R is the f -capacity of the annulus B R \ B r . Proof. Since Δ f u does not change the sign on M, Considering u is bounded from above and u > 0, we shall find a positive constant C such that u 2 ≤ C on M. For a geodesic ball B R of radius R around p ∈ M, by Lemma 6, for any r such that 0 < r < R, we have that the function u satisfies Taking into account that M is f -parabolic, we know that 1/μ r,R ⟶ 0 as R ⟶ ∞, that is, j∇uj 2 vanishes identically on M. So, u is constant on M.
On the other hand, when Δ f u ≤ 0, it follows that u is a f -superharmonic function on M, which is bounded from above. So, the conclusion now follows from f -parabolicity.
In the following, applying the weak maximum principle, we provide the sign relationship between the f -mean curvature and the derivative of warping function, in which the results extend the Lemma 14 in [15]. We point out that, to prove the following results, we do not require that the f -mean curvature H f of the hypersurface Σ n is constant.
First, recall that a slab of a weighted Riemannian warped product I × ρ M n f is a region of the type Proof. Since the hypersurface Σ n is contained in a slab, then the height function h is bounded and sup Σ σðhÞ = σðh * Þ, inf Σ σðhÞ = σðh * Þ, where h * = sup Σ h, h * = inf Σ h. Applying the weak maximum principle to the f -Laplacian Δ f σðhÞ, we may find two sequences fp j g, fq j g ⊂ Σ n such that From (11), we have Δ f σðhÞ = nρðhÞððρ ′ /ρÞðhÞ + H f ΘÞ, then where the last inequality is due to −1 ≤ Θ ≤ 0. Furthermore, taking into account that ρðhÞ is monotonic, therefore ρ′ðhÞ ≥ 0.

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Now assume that H f < 0, from (21), we have Therefore, we conclude that So ρ′ðhÞ ≤ 0.
After the following theorem, we derive our uniqueness results for parabolic hypersurfaces. we obtain H f ≤ ðρ′/ρÞðhÞ. Therefore, from (11), we have where the first inequality is due to Θ ≥ −1. Moreover, since σðhÞ is a positive smooth function and there is a constant C such that σðhÞ ≤ C. From Proposition 7, we conclude that σðhÞ, and hence, h is constant. Consequently, Σ n is a slice.
Finally, in the case where H f < 0, we know from Proposition 8 that ρ ′ ðhÞ ≤ 0, so that H f ≥ ðρ ′ /ρÞðhÞ. Therefore, The proof then follows as in the case H f > 0. Moreover, if warping function ρðhÞ satisfies condition (ii), then using (13), we have the next result which extends Theorem 9. Proof. From (13), we have By the hypothesis, we have Δ f ρðhÞ ≥ 0. Moreover, since Σ n lies in a slab, and ρðhÞ is a positive smooth function, then there exists a positive constant C such that ρðhÞ ≤ C. So, we can apply Proposition 7 to get ρðhÞ as constant. Therefore, Σ n is a slice. Now, we consider the ðn + 1Þ-dimensional weighted hyperbolic space ℍ n+1 , which instead of the more commonly used weighted half-space model, as the weighted warped product ℝ × e t ℝ n f . It can be easily seen that the slices ft 0 g × ℝ n f of ℍ n+1 = ℝ × e t ℝ n f are precisely the horospheres. Furthermore, according to Theorem 10, we have the following application in weighted hyperbolic space. Corollary 11. Let ℍ n+1 = ℝ × e t ℝ n f be a weighted hyperbolic space whose fiber ℝ n has f -parabolic universal covering and let ψ : Σ n ⟶ ℍ n+1 be a complete two-sided hypersurface which is contained in a slab. If H 2 f ≤ ð1/Θ 2 Þ, then Σ n is a slice.
Next, we will use the weak maximum principle to study another rigidity of the hypersurfaces in weighted Riemannian warped products.

Theorem 12. Let
M n+1 = I × ρ M n f be a weighted Riemannian warped product whose fiber M has f -parabolic universal covering. Let ψ : Σ n ⟶ M n+1 be a complete two-sided hypersurface which lies in a slab. Suppose the warping function ρðhÞ satisfies condition (ii), and there is a point h 0 ∈ I such that ρ ′ ðh 0 Þ = 0. If H f does not change sign, then H f = 0 and Σ n is a slice.
Proof. Since the hypersurface Σ n is contained in a slab, then h is bounded, and sup Σ h = h * , inf Σ h = h * . Reasoning as in the proof of Theorem 9, we have the weak maximum principle for f -Laplace operator Δ f holds on Σ n ; then, there exist two