# New Bernstein Type Results in Weighted Warped Products

Authors: Ning Zhang
Journal: Journal of Mathematics
 (2021)

---

## Abstract

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.


---

## Body

## 1. Introduction

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 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 productI×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−fdM, 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(1)Ricf=Ric+Hessf.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 manifoldMf and proved mean curvature and volume comparison results under the assumption that Ricf 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×ρMfn 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 Section2, 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 [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 (Theorem 2).


## 2. Preliminaries

LetMn be a connected n≥2-dimensional oriented Riemannian manifold and I⊂ℝ be an open interval endowed with the metric dt2. Let ρ:I⟶ℝ+ be a smooth function. Denote by I×ρMn the warped product manifold with the Riemannian metric(2),=πI∗dt2+ρπI2πM∗,M,where πI and πM are the projections onto I and M, respectively. This resulting space is a warped product in the sense of [6], with baseI,dt2, fiberM,,M, and warping functionρ. Furthermore, for every t0∈I, we say that Mt0n=t0×Mn is a slice of I×ρMn.Consider the vector fieldK=ρπI∂t in I×ρMn, where ∂t=∂/∂t is the unit vector field tangent to base I. Moreover, using the relationship between the Levi-Civita connections of I×ρMn and those of the fiber and the base (see Corollary 7.35 in [6]), we have(3)∇¯XK=ρ′πIX,for any X∈XI×ρMn, where ∇¯ stands for the Levi-Civita connection of the Riemannian metric , in (2). Therefore, K is conformal with ℒK,=2ρ′πI, and its metrically equivalent 1-form is closed.Recall that a smooth immersionψ:Σn⟶I×ρMn of an n-dimensional connected manifold Σn is said to be a hypersurface. Moreover, the induced metric via ψ on Σn will be also denoted by ,.In this paper, we study the connected hypersurfacesψ:Σn⟶I×ρMn oriented a unit normal vector field N. Let ∇ be the Levi-Civita connection of Σn. The Gauss and Weingarten formulas for the hypersurfaces ψ:Σn⟶I×ρMn are given, respectively, by(4)∇¯XY=∇XY+AX,YN,(5)AX=−∇¯XN,where X,Y∈XΣn and A:XΣn⟶XΣn is the shape operator (or Weingarten endomorphism) of Σn with respect to N.In the following, we consider two particular functions naturally attached to hypersurfaceΣn, namely, the angle (or support) functionΘ=N,∂t and the height functionh=πI|Σn.Let∇¯ and ∇ be the gradients with respect to the metrics of I×ρMn and Σn, respectively. Then, by a simple computation, we obtain(6)∇¯πI=∇¯πI,∂t∂t=∂t.So, the gradient ofh on Σn is(7)∇h=∇¯πI⊤=∂t⊤=∂t−ΘN.Particularly,(8)∇h2=1−Θ2,where ⋅ denotes the norm of a vector field on Σn.Moreover, taking tangential components in (3), we have, from (4) and (5), that(9)∇XK⊤=ρhAΘX+ρ′hX,where ρ′h=ρ′∘h and K⊤=ρh∂t⊤=K−K,NN is the tangential component of K along Σn. This enables us to use (9) to compute the gradient of the angle function Θ, obtaining(10)∇Θ=−A∇h−ρ′hρhΘ∇h.Furthermore, it follows from (7) and (9) that the Laplacian of h on Σn is(11)Δh=logρ′hn−∇h2+nHΘ.Consequently, byΔρh=ρ′hΔh+ρ″h∇h2, we have(12)Δρh=nρ′h2ρh+ρhlogρ″h∇h2+nρ′hHΘ.Furthermore, consider that a warped productI×ρMn endowed with a weight function f, which will be called a weighted warped product I×ρMfn. In this setting, for a hypersurface Σn immersed into I×ρMfn, the f-divergence operator on Σn is defined by(13)divfX=efdive−fX,where X is a tangent vector field on Σn.For a smooth functionu:Σn⟶ℝ, we define its drifting Laplacian by(14)Δfu=divf∇u=Δu−∇u,∇f,and we will also denote such an operator as the f-Laplacian of Σn.According to Gromov [14], the weighted mean curvature or f-mean curvature Hf of Σn is given by(15)nHf=nH+∇¯f,N,where H is the standard mean curvature of hypersurface Σn with respect to N.In this paper, we will consider weighted Riemannian warped productsI×ρMfn whose weight function f does not depend on the parameter t∈I, that is, ∇¯f,∂t=0. Moreover, we will refer to them as M¯n+1:=I×ρMfn.


## 3. Parametric Uniqueness Results

In order to prove our uniqueness results in weighted warped productM¯n+1, we need a few previous results.Lemma 1.
Letψ:Σn⟶M¯n+1 be a hypersurface in a weighted Riemannian warped product M¯n+1=I×ρMfn. Then, the height function h of Σn satisfies(16)12Δf∇h2=nΘ∇Hf,∂t+nρ′hρhHfΘ1+Θ2−4ρ′hρhΘA∇h,∇h+ρ′h2ρh2Θ2n−3|∇h|2+Θ2A2−A∇h2+Θ2RicfMN∗,N∗+nlogρ″h∇h2,where RicfM stands for the Bakry–Émery–Ricci curvature tensor of Mn and N∗=N−N,∂t∂t is the projection of the vector field N onto Mn.Proof.
The key idea of the proof is to compute thef-Laplacian of the function Θ. To do so, taking into account (14), it follows that(17)ΔfρhΘ=ΔρhΘ−∇ρhΘ,∇f.
Proceeding as above in Section2, by a direct computation from (10) and (12), gives(18)ΔρhΘ=ΘΔρh+ρhΔΘ+2∇ρh,∇Θ=nρ′h2ρhΘ+ρhlogρ″hΘ∇h2−2ρ′hA∇h,∇h−2ρ′h2ρhΘ∇h2+nHρ′hΘ2+ρhΔΘ.
Moreover, from a straightforward computation, we obtain(19)∇ρhΘ,∇f=−ρ′hΘ2∇¯f,N+ρh∇Θ,∇f.
So, by (14) and (15), we can write (17) as(20)ΔfρhΘ=nρ′h2ρhΘ+ρhlogρ″hΘ∇h2−2ρ′hA∇h,∇h−2ρ′h2ρhΘ∇h2+nρ′hΘ2Hf+ρhΔfΘ.
On the contrary, Lemma1 in [11] proves that(21)ΔfρhΘ=−nρh∇Hf,∂t−nρ′hHf−ρhΘA2−ρhΘRicMN∗,N∗+n−1logρ″h∇h2−ρhΘHess¯fN,N,where RicM stands for the Ricci curvature tensor of Mn.
Moreover, taking into account that∇¯f,∂t=0, it is easy to obtain that(22)Hess¯fN,N=HessMfN∗,N∗.
Hence, using relation (1), the result follows from (20)–(22) that(23)ΔfΘ=−n∇Hf,∂t−nρ′hρhHf1+Θ2+2ρ′hρhA∇h,∇h−ΘRicfMN∗,N∗+nlogρ″h∇h2−ρ′h2ρh2Θn−2∇h2−ΘA2.
Moreover, from (10), we obtain that(24)∇Θ2=ρ′h2ρh2Θ2∇h2+A∇h2+2ρ′hρhΘA∇h,∇h.
To conclude the proof, we should notice that(25)12Δf∇h2=12Δf1−Θ2=−ΘΔfΘ−∇Θ2.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.Lemma 2.
LetM,,M,e−fdM be a f-parabolic weighted manifold and u∈C2M be a positive function on M and u∗=supMu<+∞. If Δfu does not change sign on M, then u is constant on M.Moreover, we also need to consider the weighted warped productsI×ρMfn satisfy the following convergence condition:(26)RicM≥n−1supIρ′2−ρ″ρ,M.Theorem 1.
Letψ:Σn⟶I×ρMfn be a complete f-parabolic hypersurface with constant f-mean curvature Hf in a weighted warped product I×ρMfn which satisfies the convergence condition (26) and the weight function f is convex. Assume that the warping function satisfies ρ″h≥0. If either −1≤Θ<−3/2 and Hfρ′h≤0 or3/2<Θ≤1 and Hfρ′h≥0, then Σn is totally geodesic. In addition, if inequality (26) is strict or ρ″h>0, then Σn is a totally geodesic minimal slice.Proof.
From (1) and (22), we deduce that(27)RicfMN∗,N∗=RicMN∗,N∗+Hess¯fN,N.
Moreover, using Young’s inequality, we have(28)−4ρ′hρhΘA∇h,∇h≥−2ρ′h2ρh2Θ2∇h2−2A∇h2.
So, we can estimate(29)−4ρ′hρhΘA∇h,∇h+ρ′h2ρh2Θ2n−4∇h2+Θ2A2−A∇h2≥ρ′h2ρh2Θ2n−6∇h2+Θ2A2−3A∇h2≥ρ′h2ρh2Θ2n−6+6Θ2+4Θ2−3A2.
Considering the assumptions of Theorem1, it follows from (16) that(30)12Δf∇h2≥ρ′h2ρh2Θ2n−6+6Θ2+4Θ2−3A2+nρ′hρhHfΘ1+Θ2≥0.
Moreover,∇h2 is bounded on Σn which allows us to apply Lemma 2 to guarantee that ∇h2 is constant. So, Δf∇h2=0. From (16), we also have that(31)nρ′hρhHfΘ1+Θ2+ρ″ρΘ2∇h2+ρ′h2ρh2Θ2n−6+6Θ2+4Θ2−3A2+Θ2RicMN∗,N∗+n−1logρ″ρh∇h2=0.
SinceΘ2>3/4, we get ρ′h=0andA=0.
So,Σn is totally geodesic in I×ρMfn with ρ=c=constant. Finally, when inequality (26) is strict, it follows from (31) that N∗p=0 at any p∈Σn, that is, ∇h=0 on Σn. Therefore, Σn is a slice.
Moreover, ifρ″h>0, (31) implies that ∇h2=0 on Σn; consequently, h is constant, that is, Σn is a slice.Remark 1.
Notice that, in the case wheref-weighted mean curvature Hf≡0, i.e., Σn is a f-minimal hypersurface, we reobtain Theorem 6 in [11]. Thus, Theorem 1 extends this previous result to the constant f-weighted mean curvature case in weighted warped products.
It should also note that the statement of Theorem1 could be improved by requiring that(32)RicfM≥n−1supIρ′2−ρ″ρ,Minstead of separately requiring the convergence condition (26), and the weight function f is convex, i.e.,(33)RicM≥n−1supIρ′2−ρ″ρ,M,and Hessf≥0.


## 4. Bernstein Type Results

In this section, we will consider the case of a vertical graph over the fiberMn,,M in a weighted warped product M¯n+1=I×ρMfn, n≥2, defined by(34)Σnu=ux,x:x∈Ω⊂M¯n+1,where Ω⊂Mn is a connected domain of Mn and u is a smooth function on Ω. Moreover, the metric induced on Ω from the metric on M¯n+1 via identification with Σnu is given by(35),=du2+ρu2,M.It is easy to see from this expression that ifMn is complete and infΣnuρu>0, then the graph Σnu is complete. Furthermore, the graph Σnu is said to be entire if Ω=Mn. On the contrary, we note that hup,p=up for any point p∈Mn, so h and u may be naturally identified on Σnu.WhenΣnu is entire, the unitary normal vector field of Σnu in M¯n+1=I×ρMfn is given by(36)N=1ρuρu2+Du2ρu2∂t−Du,where Du is the gradient of u on Mn and Du2=Du,DuM. Using Proposition 7.35 in [6] again, we can get the shape operator associated to N:(37)AX=1ρu2+Du2ρ′uX−1ρuDXDu+DXDu,DuMρuρu2+Du23/2+ρ′uDu,XMρu2+Du23/2Du,for X∈XΩ, where D denotes the Levi-Civita connection in Mn. Consequently, it is easy to verify from (15) and (37) that the f-mean curvature Hf of Σnu corresponding to N is given by(38)Hfu=divfDunρuρu2+Du2−ρ′unρu2+Du2n−Du2ρu2.In particular, an entire graphΣnu has constant f-mean curvature if and only if the function u∈C∞M satisfies the following elliptic partial differential equation:(39)divfDuρuρu2+Du2−ρ′uρu2+Du2n−Du2ρu2=C,for some constant C∈ℝ.In what follows, we will use the results of Section3 to obtain a new uniqueness result for equation (39).Theorem 2.
LetM¯n+1=I×ρMfn be a weighted warped product with complete fiber Mn, obeying (26). Assume that the weighted function f is convex and the warping function satisfies ρ″h≥0. Let Σu be an f-parabolic entire graph in M¯n+1 determined by a function u∈C∞M with constant f-mean curvature such that Hfρ′hΘ≥0. Suppose that either inequality (26) is strict or ρ″h>0, then, for constant 0≤λ≤3/3, the only bounded entire solutions to equation (39) with Du<λρu are the constant ones u=t0 for some t0∈I.Proof.
Note that, by a straightforward computation, we can get the following relation:(40)∇h2=1−Θ2=Du2ρu2+Du2.
Therefore, by the constraintsDu<λρu, 0≤λ≤3/3, and (40), we have(41)11+λ2<Θ2≤1.
The rest of our assumptions enable us to apply Theorem1 to end the proof.



---

*Source: 5540712-2021-04-20.xml*