Advances in Mathematical Physics

Advances in Mathematical Physics / 2021 / Article

Research Article | Open Access

Volume 2021 |Article ID 3234263 | https://doi.org/10.1155/2021/3234263

Ning Zhang, "Uniqueness of Complete Hypersurfaces in Weighted Riemannian Warped Products", Advances in Mathematical Physics, vol. 2021, Article ID 3234263, 9 pages, 2021. https://doi.org/10.1155/2021/3234263

Uniqueness of Complete Hypersurfaces in Weighted Riemannian Warped Products

Academic Editor: Leopoldo Greco
Received06 Jul 2020
Revised18 Dec 2020
Accepted22 Dec 2020
Published02 Jan 2021

Abstract

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 whose fiber has -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.

1. 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 is associated with a complete -dimensional Riemannian manifold , and a smooth function on is the triple , where stands for the volume element of . 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 , 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 -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 or 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 two-sided 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 where is an open interval, is a complete -dimensional Riemannian manifold, is a positive smooth warping function, and the warped metric is given by where is the metric tensor of . 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 -parabolicity of weighted manifolds and pay attention to show the weak maximum principle for the -Laplace operator holds on -parabolic weighted manifolds. Moreover, by using the weak maximum principle, we provide the sign relationship among the -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 -mean curvature and the warping function in weighted Riemannian warped products whose fiber has -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.

2. Preliminaries

Let be a connected -dimensional oriented Riemannian manifold and be an open interval which is endowed with the metric . Let be a positive smooth function. Denote to be the product manifold with the following Riemannian metric

where and are the projections onto and , respectively. Following the terminology used in [7], Chap.7, this resulting space is a warped product with fiber, base, and warping function. Furthermore, for a fixed point , we say that is a slice of .

Recalling that a smooth immersion of an -dimensional connected manifold is called to be a hypersurface. Moreover, the induced metric via on will be also denoted for .

Throughout this paper, we assume that is a two-sided hypersurface. Recalling that a hypersurface is called a two-sided hypersurface if its normal bundle is trivial, which means that there exists a globally defined unit normal vector field . For instance, every hypersurface with never vanishing mean curvature is trivially two-sided. Moreover, when the hypersurface is two-sided, a choice of on makes the second fundamental form globally defined on . In the sequel, the Riemannian warped product is clearly orientable. This allows us to take, for each two-sided hypersurface , a unique unitary normal vector field globally defined on in the same orientation of the vector field , , i.e., such that . By the wrong-way Cauchy-Schwarz inequality (see [7], Proposition 5.30), we have , and the first equality holds at a point if and only if at . Moreover, we will refer to the function , , as the angle function. On the other hand, we will represent a particular function naturally attached to the hypersurface by the height function.

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 of has constant mean curvature with respect to the unit normal vector field .

Let and stand for gradients with respect to the metrics of and , respectively. In a simple computation, we have

So, the gradient of on is

Particularly, we have

where || denotes the norm of a vector field on .

Now, we consider that a Riemannian warped product endowed with a weighted function , which will be called a weighted Riemannian warped product . In this setting, for a two-sided hypersurface immersed into , the -divergence operator on is defined by where is a tangent vector field on .

For a smooth function , we define its drifting Laplacian by

we will also denote such an operator as the -Laplacian of .

According to Gromov [8], the weighted mean curvature, or -mean curvature of , is given by where is the standard mean curvature of hypersurface with respect to the Gauss map .

Notice it follows from a splitting theorem by the case (see [9], Theorem 1.2) that if a weighted Riemannian warped product is endowed with a bounded weighted function and such that for all vector fields on , then must be constant along . So, motivated by this result, in the following, we will consider weighted Riemannian warped products whose weighted function does not depend on the parameter , that is . Moreover, for simplicity, we will refer to them as .

Remark 1. We note that the -mean curvature of a slice in a weighted Riemannian warped product is given by,

Indeed, since is a normal vector field to the slice , from (9), we have that .

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 be a hypersurface immersed in a weighted Riemannian warped product, with height function . Then where is a primitive function of .

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].

Lemma 3 ([11]). Let be an -dimensional (not necessarily complete) weighted Riemannian manifold. We say that the weak maximum principle for the -Laplace operator holds on , if for any smooth bounded above function on , there exists a sequence such that

Equivalently, for any smooth bounded below function on , then there is a sequence such that

On the other hand, a smooth function on a weighted manifold is called -superharmonic if . Taking this into account, a noncompact weighted manifold is said to be -parabolic if it does not admit nonconstant positive -superharmonic functions on . So, we can conclude the following extension of Theorem 1 in [12], which establishes sufficient conditions to ensure that the two-sided hypersurface in is -parabolic.

Lemma 4 ([12]). Let be a complete two-sided hypersurface in a weighted Riemannian warped product whose fiber has -parabolic universal covering. If the angle function is bounded and the restriction on of warping function satisfies:
(c1)
(c2)
then, is -parabolic.

3. Uniqueness Results in Weighted Riemannian Warped Products

In this section, we will study the uniqueness for complete hypersurfaces in weighted Riemannian warped products . Before describing our main results, we will prove some auxiliary propositions which will be essential in the sequel.

Proposition 5. If the weighted manifold is -parabolic, then the weak maximum principle for the -Laplace operator holds on .

Proof. Since weighted manifold is -parabolic, using Corollary 6.4 in [13] it follows that is also stochastically complete.

On the other hand, by the fact which in [11] that satisfies the weak maximum principle for the -Laplace operator if and only if is stochastically complete, this concludes the proof.

Furthermore, for any compact subset , we define the -capacity of as, where is the set of all compactly supported Lipschitz functions on . By the fact that a weighted manifold is -parabolic if and only if for any compact set .

The following lemma is the extension of Lemma 3 in [14], which will allow us to obtain our technical result.

Lemma 6 ([14]). Let be an -dimensional weighted manifold and consider which satisfies . Let be a geodesic ball of radius around . For any such that , we have where denotes the geodesic ball of radius around and is the -capacity of the annulus .

Proposition 7. Let be an -parabolic weighted manifold and be a positive function on and . If does not change the sign on , then is constant on .

Proof. Since does not change the sign on , thus or . If , then . Considering is bounded from above and , we shall find a positive constant such that on . For a geodesic ball of radius around , by Lemma 6, for any such that , we have that the function satisfies

Taking into account that is -parabolic, we know that as , that is, vanishes identically on . So, is constant on .

On the other hand, when , it follows that is a -superharmonic function on , which is bounded from above. So, the conclusion now follows from -parabolicity.

In the following, applying the weak maximum principle, we provide the sign relationship between the -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 -mean curvature of the hypersurface is constant.

First, recall that a slab of a weighted Riemannian warped product is a region of the type

Proposition 8. Let be a hypersurface with nonvanishing -mean curvature which is contained in a slab. Choose on the orientation such that . Assume that the weak maximum principle for the -Laplace operator holds on . If either (i)The warping function is monotonic or(ii)The function is nondecreasingthen . On the other hand, if , then .

Proof. Since the hypersurface is contained in a slab, then the height function is bounded and , where , . Applying the weak maximum principle to the -Laplacian , we may find two sequences such that

From (11), we have , then (i)Since , from (22), we get

where the last inequality is due to . Furthermore, taking into account that is monotonic, therefore .

On the other hand, since and , jointly with (21), we have

So follows from that is monotonic. (ii)Since , , and is nondecreasing, it follows from (22) that

Therefore,

So .

Now assume that , from (21), we have

Therefore, we conclude that

So .

After the following theorem, we derive our uniqueness results for parabolic hypersurfaces.

Theorem 9. Let be a weighted Riemannian warped product whose fiber has -parabolic universal covering. Let be a complete hypersurface with nonvanishing -mean curvature which lies in a slab. Suppose the warping function satisfies conditions or . If , then is a slice.

Proof. Since has -parabolic universal covering, lies in a slab and ; then, we deduce that is -parabolic by Lemma 4. Moreover, from Proposition 5, it follows that weak maximum principle for the -Laplace operator holds on . Proceeding as above and considering the assumption that the warping function satisfies conditions (i) or (ii), we have that Proposition 8 holds true.

In the case where , by Proposition 8, we have . Combining the assumption , we obtain . Therefore, from (11), we have

where the first inequality is due to .

Moreover, since is a positive smooth function and there is a constant such that . From Proposition 7, we conclude that , and hence, is constant. Consequently, is a slice.

Finally, in the case where , we know from Proposition 8 that , so that . Therefore,

The proof then follows as in the case .

Moreover, if warping function satisfies condition (ii), then using (13), we have the next result which extends Theorem 9.

Theorem 10. Let be a weighted Riemannian warped product whose fiber has -parabolic universal covering. Let be a complete two-sided hypersurface that lies in a slab. Suppose the warping function satisfies conditions (ii). If the -mean curvature satisfies , then is a slice.

Proof. From (13), we have

By the hypothesis, we have . Moreover, since lies in a slab, and is a positive smooth function, then there exists a positive constant such that . So, we can apply Proposition 7 to get as constant. Therefore, is a slice.

Now, we consider the -dimensional weighted hyperbolic space , which instead of the more commonly used weighted half-space model, as the weighted warped product . It can be easily seen that the slices of are precisely the horospheres. Furthermore, according to Theorem 10, we have the following application in weighted hyperbolic space.

Corollary 11. Let be a weighted hyperbolic space whose fiber has -parabolic universal covering and let be a complete two-sided hypersurface which is contained in a slab. If , then 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 be a weighted Riemannian warped product whose fiber has -parabolic universal covering. Let be a complete two-sided hypersurface which lies in a slab. Suppose the warping function satisfies condition (ii), and there is a point such that . If does not change sign, then and is a slice.

Proof. Since the hypersurface is contained in a slab, then is bounded, and . Reasoning as in the proof of Theorem 9, we have the weak maximum principle for -Laplace operator holds on ; then, there exist two sequences such that

From (12), we have that

Take . Thus, it follows from (33) and (34) that and

Furthermore, taking into account that is nondecreasing yields

On the other hand, by (6), we have . Therefore, using (35) and (36), we conclude that

Considering that does not change the sign on , hence , that is, is a -minimal hypersurface. Using (13), we have

In the following, by the same argument as in Theorem 10, we have is a slice.

4. Uniqueness Results in Weighted Product Spaces

In this section, we establish some uniqueness results concerning the complete hypersurfaces in weighted product spaces . Firstly, as a consequence of Theorem 9, when the ambient space is weighted product space, we have the following result.

Corollary 13. Let be a weighted product space whose fiber has -parabolic universal covering, and let be a complete hypersurface with nonvanishing -mean curvature which is contained in a slab. If does not change the sign on , then is a slice.

Proof. From (12), we have

Since does not change sign, we have does not change the sign on , and proceeding as in the proof of Theorem 9, we obtain that is parabolic. Furthermore, we know that height function is bounded, which implies that is a slice from Proposition 7.

Theorem 14. Let be a complete hypersurface with nonvanishing constant -mean curvature in a weighted product space whose fiber has -parabolic universal covering. Assume that for some nonnegative constant and the weighted function is convex. If , then is a slice.

Proof. Let be a (local) orthonormal frame in ; using the Gauss equation, we have for any . Moreover, we also have where is the sectional curvature of , and are the projections of the tangent vector fields and onto , respectively.

Considering the hypothesis that for some nonnegative constant , and by direct computation, we have

Summing up,

Furthermore, taking into account that the weighted function is convex, we have

So

Particularly,

On the other hand, using the Bochner-Lichnerowiz formula (see [2]), where

Thus, from (5), we have that

Using as a constant and (12), we have

Consequently,

Next, by the hypothesis and (6), we have . So, from (52), it follows that

Moreover, reasoning as in the proof of Theorem 9, we have that is parabolic. Since is bounded on , thus by Proposition 7, we conclude that is constant. So, . From (53), we also have on ; consequently, is constant, that is, is a slice.

To prove our next result, we need the following auxiliary lemma.

Lemma 15 ([3]). Let be a hypersurface with constant -mean curvature immersed in a weighted product space ; then where is the Bakry-Émery-Ricci tensor of the fiber and is the orthonormal projection of onto .

Theorem 16. Let be a complete hypersurface with nonvanishing constant -mean curvature in a weighted product space whose fiber has -parabolic universal covering. Assume that for some positive constant and the weighted function is convex. If , for some constant , then is a slice.

Proof. By the Gauss equation and with a direct computation, we have

In the case where , from Lemma 15 and the condition , we have

On the other hand, as we did before in the proof of Theorem 9, it follows that is parabolic. Moreover, since is bounded on , we conclude from Proposition 7 that is constant. So, . Using (56), we have , which implies that is constant. Therefore, is a slice.

As a consequence of the proof of Theorem 16, we can get the following corollary.

Corollary 17. Let be a complete hypersurface with nonvanishing constant -mean curvature in a weighted product space whose fiber has -parabolic universal covering. Assume that and for some positive constants and . If , for some constant , then is a slice.

5. Nonparametric Results for the Entire Graphs

In this section, we consider the vertical graphs in a weighted Riemannian warped product , which are defined by where be a connected domain of and is a smooth function on . Moreover, the metric induced on from the metric on ambient space via , which is represented by

It is easy to see from the metric induced on of that if the function is bounded on , the graphs is complete. Furthermore, the graph is said to be entire if .

In the following, we can give the reason as in the proof of the Theorem 9 to obtain a nonparametric result.

Corollary 18. Let be a weighted Riemannian warped product whose fiber has -parabolic universal covering, and let be an entire graph with nonvanishing -mean curvature which lies in a slab. Suppose the warping function satisfies condition or . If , then is a slice.

Next, it is not difficult to obtain the following nonparametric version of Theorem 10.

Corollary 19. Let be a weighted Riemannian warped product whose fiber has -parabolic universal covering. Let be an entire graph which lies in a slab. Suppose the warping function satisfies condition (ii). If the -mean curvature satisfies , then is a slice.

If, moreover, , we can have the following corollaries of all other theorems of Section 4.

Corollary 20. Let be an entire graph with nonvanishing constant -mean curvature in a weighted product space whose fiber has -parabolic universal covering. Assume that for some nonnegative constant and the weighted function is convex. If , then is a slice.

Corollary 21. Let be an entire graph with nonvanishing constant -mean curvature in a weighted product space whose fiber has -parabolic universal covering. Assume that for some positive constant and the weighted function is convex. If , for some constant , then is a slice.

Data Availability

No data used to support this study.

Conflicts of Interest

The author declares that there is no conflict of interest regarding the publication of this paper.

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Copyright © 2021 Ning Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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