New Bernstein Type Results in Weighted Warped Products
In this paper, we obtain new parametric uniqueness results for complete constant weighted mean curvature hypersurfaces under suitable geometric assumptions in weighted warped products. Furthermore, we also prove very general Bernstein type results for the constant mean curvature equation for entire graphs in these ambient spaces.
In the last decades, constant mean curvature hypersurfaces in Riemannian manifolds have been deeply studied. This is because that such hypersurfaces exhibit nice Bernstein type properties. Indeed, the classical Bernstein theorem states that a complete minimal surface in is a plane. Later, the Bernstein theorem has been extended to higher dimensions as follows: each complete minimal hypersurface in must be hyperplane by Simons in  for , and the result was achieved through successive efforts of Almgren , Fleming  and De Giorgi . However, for , there exists a counterexample in , by constructing a nontrivial complete minimal hypersurface in , 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 , in the sense of , with an interval equipped with a positive definite metric as base, a Riemannian manifold as fiber, and a positive smooth function 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 associated with a complete Riemannian manifold and a smooth positive function on is the triple , where is the volume element of . In this setting, we will consider the Bakry–Émery–Ricci tensor (see ) which is a generalization of the standard Ricci tensor 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 and proved mean curvature and volume comparison results under the assumption that is bounded from below and or is bounded in . In particular, Salamanca and Salavessa  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.  obtained Liouville type results for two-sided hypersurfaces in weighted Killing warped products. More recently, the author  proved some uniqueness results of complete hypersurfaces in weighted Riemannian warped products with -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 (Theorem 1) which extend the corresponding results in . 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 (Theorem 2).
Let be a connected -dimensional oriented Riemannian manifold and be an open interval endowed with the metric . Let be a smooth function. Denote by the warped product manifold with the Riemannian metricwhere and are the projections onto and , respectively. This resulting space is a warped product in the sense of , with base, fiber, and warping function. Furthermore, for every , we say that is a slice of .
Consider the vector field in , where is the unit vector field tangent to base . Moreover, using the relationship between the Levi-Civita connections of and those of the fiber and the base (see Corollary 7.35 in ), we havefor any , where stands for the Levi-Civita connection of the Riemannian metric in (2). Therefore, is conformal with and its metrically equivalent 1-form is closed.
Recall that a smooth immersion of an -dimensional connected manifold is said to be a hypersurface. Moreover, the induced metric via on will be also denoted by .
In this paper, we study the connected hypersurfaces oriented a unit normal vector field . Let be the Levi-Civita connection of . The Gauss and Weingarten formulas for the hypersurfaces are given, respectively, bywhere and is the shape operator (or Weingarten endomorphism) of with respect to .
In the following, we consider two particular functions naturally attached to hypersurface , namely, the angle (or support) function and the height function.
Let and be the gradients with respect to the metrics of and , respectively. Then, by a simple computation, we obtain
So, the gradient of on is
Particularly,where denotes the norm of a vector field on .
Moreover, taking tangential components in (3), we have, from (4) and (5), thatwhere and is the tangential component of along . This enables us to use (9) to compute the gradient of the angle function , obtaining
Furthermore, it follows from (7) and (9) that the Laplacian of on is
Consequently, by , we have
Furthermore, consider that a warped product endowed with a weight function , which will be called a weighted warped product . In this setting, for a hypersurface immersed into , the -divergence operator on is defined bywhere is a tangent vector field on .
For a smooth function , we define its drifting Laplacian byand we will also denote such an operator as the -Laplacian of .
According to Gromov , the weighted mean curvature or -mean curvature of is given bywhere is the standard mean curvature of hypersurface with respect to .
In this paper, we will consider weighted Riemannian warped products whose weight function does not depend on the parameter , that is, . Moreover, we will refer to them as .
3. Parametric Uniqueness Results
In order to prove our uniqueness results in weighted warped product , we need a few previous results.
Lemma 1. Let be a hypersurface in a weighted Riemannian warped product . Then, the height function of satisfieswhere stands for the Bakry–Émery–Ricci curvature tensor of and is the projection of the vector field onto .
Proof. The key idea of the proof is to compute the -Laplacian of the function . To do so, taking into account (14), it follows thatProceeding as above in Section 2, by a direct computation from (10) and (12), givesMoreover, from a straightforward computation, we obtainSo, by (14) and (15), we can write (17) asOn the contrary, Lemma 1 in  proves thatwhere stands for the Ricci curvature tensor of .
Moreover, taking into account that , it is easy to obtain thatHence, using relation (1), the result follows from (20)–(22) thatMoreover, from (10), we obtain thatTo conclude the proof, we should notice thatand use (23) and (24) to obtain (16).
In the following, we give the next technical lemma (for further details, see Proposition 3.3 in ) which will be essential for the proofs of our main results.
Lemma 2. Let be a -parabolic weighted manifold and be a positive function on and . If does not change sign on , then is constant on .
Moreover, we also need to consider the weighted warped products satisfy the following convergence condition:
Theorem 1. Let be a complete -parabolic hypersurface with constant -mean curvature in a weighted warped product which satisfies the convergence condition (26) and the weight function is convex. Assume that the warping function satisfies . If either and or and , then is totally geodesic. In addition, if inequality (26) is strict or , then is a totally geodesic minimal slice.
Proof. From (1) and (22), we deduce thatMoreover, using Young’s inequality, we haveSo, we can estimateConsidering the assumptions of Theorem 1, it follows from (16) thatMoreover, is bounded on which allows us to apply Lemma 2 to guarantee that is constant. So, . From (16), we also have thatSince , we get .
So, is totally geodesic in with . Finally, when inequality (26) is strict, it follows from (31) that at any , that is, on . Therefore, is a slice.
Moreover, if , (31) implies that on ; consequently, is constant, that is, is a slice.
Remark 1. Notice that, in the case where -weighted mean curvature , i.e., is a -minimal hypersurface, we reobtain Theorem 6 in . Thus, Theorem 1 extends this previous result to the constant -weighted mean curvature case in weighted warped products.
It should also note that the statement of Theorem 1 could be improved by requiring thatinstead of separately requiring the convergence condition (26), and the weight function is convex, i.e.,
4. Bernstein Type Results
In this section, we will consider the case of a vertical graph over the fiber in a weighted warped product , , defined bywhere is a connected domain of and is a smooth function on . Moreover, the metric induced on from the metric on via identification with is given by
It is easy to see from this expression that if is complete and , then the graph is complete. Furthermore, the graph is said to be entire if . On the contrary, we note that for any point , so and may be naturally identified on .
When is entire, the unitary normal vector field of in is given bywhere is the gradient of on and . Using Proposition 7.35 in  again, we can get the shape operator associated to :for , where denotes the Levi-Civita connection in . Consequently, it is easy to verify from (15) and (37) that the -mean curvature of corresponding to is given by
In particular, an entire graph has constant -mean curvature if and only if the function satisfies the following elliptic partial differential equation:for some constant .
In what follows, we will use the results of Section 3 to obtain a new uniqueness result for equation (39).
Theorem 2. Let be a weighted warped product with complete fiber , obeying (26). Assume that the weighted function is convex and the warping function satisfies . Let be an -parabolic entire graph in determined by a function with constant -mean curvature such that . Suppose that either inequality (26) is strict or , then, for constant , the only bounded entire solutions to equation (39) with are the constant ones for some .
Proof. Note that, by a straightforward computation, we can get the following relation:Therefore, by the constraints , , and (40), we haveThe rest of our assumptions enable us to apply Theorem 1 to end the proof.
No data were used to support this study.
Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.
This work was supported by the Youth Science Foundation of Henan Institute of Technology (no. KQ1906), Key Scientific Research Project of Colleges and Universities in Henan Province (no. 20B120001), and Key Scientific and Technological Project of Henan Province (no. 212102210247).
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