# Uniqueness of Complete Hypersurfaces in Weighted Riemannian Warped Products

Authors: Ning Zhang
Journal: Advances in Mathematical Physics
 (2021)

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## 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 productI×ρMfn 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.


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## Body

## 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 theweighted manifolds, which are also called manifolds with density or smooth metric measure spaces. More precisely, the weighted manifold Mfn is associated with a complete n-dimensional Riemannian manifold Mn,g, and a smooth function f on Mn is the triple Mn,g,dμ=e−fdM, where dM stands for the volume element of Mn. 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
(1)Ricf=Ric+Hessf.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 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 productI×ρMn where I⊂ℝ is an open interval, Mn is a complete n-dimensional Riemannian manifold, ρ:I⟶ℝ+ is a positive smooth warping function, and the warped metric is given by
(2),=dt2+ρt2,M,where ,M is the metric tensor of Mn. 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 Section2, 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 Δf 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×ρMfn whose fiber Mfn 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.


## 2. Preliminaries

LetMn be a connected n-dimensional oriented Riemannian manifold and I⊂ℝ be an open interval which is endowed with the metric dt2. Let ρ:I⟶ℝ+ be a positive smooth function. Denote I×ρMn to be the product manifold with the following Riemannian metric
(3),=πI∗dt2+ρπI2πM∗,M,whereπI and πM are the projections onto I and M, respectively. Following the terminology used in [7], Chap.7, this resulting space is a warped product with fiberM,,M, baseI,dt2, and warping functionρ. Furthermore, for a fixed point t0∈I, we say that Mt0n=t0×Mn is a slice of I×ρMn.Recalling that a smooth immersionψ:Σn⟶I×ρMn of an n-dimensional connected manifold Σn is called to be a hypersurface. Moreover, the induced metric via ψ on Σn will be also denoted for ,.Throughout this paper, we assume thatΣn is a two-sided hypersurface. Recalling that a hypersurface Σn is called a two-sided hypersurface if its normal bundle is trivial, which means that there exists a globally defined unit normal vector field N∈X⊥Σ. For instance, every hypersurface with never vanishing mean curvature is trivially two-sided. Moreover, when the hypersurface Σn is two-sided, a choice of N on Σn makes the second fundamental form globally defined on X⊥Σ. In the sequel, the Riemannian warped product is clearly orientable. This allows us to take, for each two-sided hypersurface Σn, a unique unitary normal vector field N globally defined on Σn in the same orientation of the vector field ∂t, ∂t≔∂/∂t, i.e., such that N,∂t≤0. By the wrong-way Cauchy-Schwarz inequality (see [7], Proposition 5.30), we have −1≤N,∂t≤0, and the first equality holds at a point p∈Σn if and only if N=−∂t at p. Moreover, we will refer to the function Θ:Σ⟶−1,0, Θ≔N,∂t, as the angle function. On the other hand, we will represent a particular function naturally attached to the hypersurface Σn by the height functionh=πIΣ:Σn⟶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 slicet0×Mn of I×ρMn has constant mean curvature H=ρ′t0/ρt0 with respect to the unit normal vector field N=−∂t.Let∇¯ and ∇ stand for gradients with respect to the metrics of I×ρMn and Σn, respectively. In a simple computation, we have
(4)∇¯πI=∇¯πI,∂t∂t=∂t.So, the gradient ofh on Σn is
(5)∇h=∇¯πIΤ=∂tΤ=∂t−ΘN.Particularly, we have(6)∇h2=1−Θ2,where || denotes the norm of a vector field onΣn.Now, we consider that a Riemannian warped productI×ρMn endowed with a weighted function f, which will be called a weighted Riemannian warped product I×ρMfn. In this setting, for a two-sided hypersurface Σn immersed into I×ρMfn, the f-divergence operator on Σn is defined by
(7)divfX=efdive−fX,where X is a tangent vector field on Σn.For a smooth functionu:Σn⟶ℝ, we define its drifting Laplacian by
(8)Δfu=divf∇u=Δu−∇u,∇f,we will also denote such an operator as thef-Laplacian of Σn.According to Gromov [8], the weighted mean curvature, or f-mean curvature Hf of Σn, is given by
(9)nHf=nH+∇¯f,N,where H is the standard mean curvature of hypersurface Σn 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×ρMfn is endowed with a bounded weighted function f and such that Ric¯fV,V≥0 for all vector fields V on I×ρMfn, then f must be constant along ℝ. So, motivated by this result, in the following, we will consider weighted Riemannian warped products I×ρMfn whose weighted function f does not depend on the parameter t∈I, that is ∇¯f,∂t=0. Moreover, for simplicity, we will refer to them as M¯n+1≔I×ρMfn.Remark 1.
We note that thef-mean curvature Hf of a slice t0×Mn in a weighted Riemannian warped product M¯n+1 is given by,
(10)Hf=ρ′t0ρt0.Indeed, since−∂t=N is a normal vector field to the slice t0×Mn, from (9), we have that Hf=H=ρ′t0/ρt0.For the proof of our main results in this paper, we need the following formulas that will be the extensions of Lemma2 in [10].Lemma 2 ([10]).
Letψ:Σn⟶I×ρMfn be a hypersurface immersed in a weighted Riemannian warped product, with height function h. Then
(11)Δfσh=nρhρ′hρh+HfΘ,(12)Δfh=logρ′hn−∇h2+nHfΘ,(13)Δfρh=nρ′h2ρh+ρhlogρ′′h∇h2+nρ′hHfΘ,
whereσt is a primitive function of ρt.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]).
LetMn,,M,e−fdM be an n-dimensional (not necessarily complete) weighted Riemannian manifold. We say that the weak maximum principle for the f-Laplace operator Δf holds on M, if for any smooth bounded above function u on M, there exists a sequence pj⊂M such that
(14)limjupj=supu,andlimjΔfupj≤0.Equivalently, for any smooth bounded below functionu on M, then there is a sequence qj⊂M such that
(15)limjuqj=infu,limjΔfuqj≥0.On the other hand, a smooth functionu on a weighted manifold Mf is called f-superharmonic if Δfu≤0. Taking this into account, a noncompact weighted manifold Mn,,,e−fdM 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 Σn in M¯n+1 is f-parabolic.Lemma 4 ([12]).
LetΣn be a complete two-sided hypersurface in a weighted Riemannian warped product I×ρMfn whose fiber M has f-parabolic universal covering. If the angle function Θ is bounded and the restriction ρh on Σn of warping function ρ satisfies:
(c1) supρh<∞
(c2) infρh>0
then,Σn is f-parabolic.


## 3. Uniqueness Results in Weighted Riemannian Warped Products

In this section, we will study the uniqueness for complete hypersurfaces in weighted Riemannian warped productsM¯n+1. Before describing our main results, we will prove some auxiliary propositions which will be essential in the sequel.Proposition 5.
If the weighted manifoldMn,,M,e−fdM is f-parabolic, then the weak maximum principle for the f-Laplace operator Δf holds on M.Proof.
Since weighted manifoldM 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 Δf if and only if M is stochastically complete, this concludes the proof.Furthermore, for any compact subsetΩ⊂Mn,,M,e−fdM, we define the f-capacity of Ω as,
(16)capfΩ=inf∫M∇u2e−fdM:u∈LipfM,uΩ≡1,where LipfM 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 capfΩ=0 for any compact set Ω.The following lemma is the extension of Lemma3 in [14], which will allow us to obtain our technical result.Lemma 6 ([14]).
LetMn,,M,e−fdM be an n≥2-dimensional weighted manifold and consider v∈C2M which satisfies vΔfv≥0. Let BR be a geodesic ball of radius R around p∈M. For any r such that 0<r<R, we have
(17)∫Br∇v2e−fdM≤4supBRv2μr,R,where Br denotes the geodesic ball of radius r around p∈M and 1/μr,R is the f-capacity of the annulus BR\B¯r.Proposition 7.
LetMn,,M,e−fdM be an f-parabolic weighted manifold and u∈C2M be a positive function on M and u∗=supMu<+∞. If Δfu does not change the sign on M, then u is constant on M.Proof.
SinceΔfu does not change the sign on M, thus Δfu≥0 or Δfu≤0. If Δfu≥0, then uΔfu≥0. Considering u is bounded from above and u>0, we shall find a positive constant C such that u2≤C on M. For a geodesic ball BR of radius R around p∈M, by Lemma 6, for any r such that 0<r<R, we have that the function u satisfies
(18)∫Br∇u2e−fdM≤4Cμr,R.Taking into account thatM is f-parabolic, we know that 1/μr,R⟶0 as R⟶∞, that is, ∇u2 vanishes identically on M. So, u is constant on M.On the other hand, whenΔfu≤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 thef-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 Hf of the hypersurface Σn is constant.First, recall that a slab of a weighted Riemannian warped productI×ρMfn is a region of the type
(19)t1,t2×Mfn=t,p∈I×ρMfn:t1≤t≤t2.Proposition 8.
Letψ:Σn⟶I×ρMfn be a hypersurface with nonvanishing f-mean curvature which is contained in a slab. Choose on Σ the orientation such that Hf>0. Assume that the weak maximum principle for the f-Laplace operator Δf holds on Σ. If either
(i)
The warping functionρ is monotonic or(ii)
The functionlogρ′ is nondecreasing
thenρ′h≥0. On the other hand, if Hf<0, then ρ′h≤0.Proof.
Since the hypersurfaceΣn is contained in a slab, then the height function h is bounded and supΣσh=σh∗,infΣσh=σh∗, where h∗=supΣh, h∗=infΣh. Applying the weak maximum principle to the f-Laplacian Δfσh, we may find two sequences pj,qj⊂Σn such that
(20)limjσhpj=σh∗,limjΔfσhpj≤0,limjσhqj=σh∗,limjΔfσhqj≥0.From (11), we have Δfσh=nρhρ′/ρh+HfΘ, then
(21)limjΔfσhpj=limjnρhpjρ′ρhpj+ΘpjHfpj≤0,(22)limjΔfσhqj=limjnρhqjρ′ρhqj+ΘqjHfqj≥0.(i)
SinceHf>0, from (22), we get(23)limjρ′ρhqj≥limj−ΘqjHfqj≥0,where the last inequality is due to−1≤Θ≤0. Furthermore, taking into account that ρh is monotonic, therefore ρ′h≥0.On the other hand, sinceHf<0 and −1≤Θ≤0, jointly with (21), we have
(24)limjρ′ρhpj≤limj−ΘpjHfpj≤0.Soρ′h≤0 follows from that ρh is monotonic.
(ii)
SinceHf>0, −1≤Θ≤0, and logρ′ is nondecreasing, it follows from (22) that(25)ρ′ρh∗≥limj−ΘqjHfqj≥0.Therefore,(26)ρ′ρh≥ρ′ρh∗≥0.Soρ′h≥0.Now assume thatHf<0, from (21), we have
(27)ρ′ρh∗≤limj−ΘpjHfpj≤0.Therefore, we conclude that(28)ρ′ρh≤ρ′ρh∗≤0.Soρ′h≤0.After the following theorem, we derive our uniqueness results for parabolic hypersurfaces.Theorem 9.
LetM¯n+1=I×ρMfn be a weighted Riemannian warped product whose fiber M has f-parabolic universal covering. Let ψ:Σn⟶M¯n+1 be a complete hypersurface with nonvanishing f-mean curvature which lies in a slab. Suppose the warping function ρh satisfies conditions i or ii. If Hf2≤ρ′2/ρ2h, then Σn is a slice.Proof.
SinceM has f-parabolic universal covering, Σ lies in a slab and −1≤Θ≤0; then, we deduce that Σ is f-parabolic by Lemma 4. Moreover, from Proposition 5, it follows that weak maximum principle for the f-Laplace operator Δf holds on Σ. Proceeding as above and considering the assumption that the warping function ρh satisfies conditions (i) or (ii), we have that Proposition 8 holds true.In the case whereHf>0, by Proposition 8, we have ρ′h≥0. Combining the assumption Hf2≤ρ′2/ρ2h, we obtain Hf≤ρ′/ρh. Therefore, from (11), we have
(29)Δfσh=nρhρ′ρh+HfΘ≥nρhρ′ρh−Hf≥0,where the first inequality is due toΘ≥−1.Moreover, sinceσh is a positive smooth function and there is a constant C such that σh≤C. From Proposition 7, we conclude that σh, and hence, h is constant. Consequently, Σn is a slice.Finally, in the case whereHf<0, we know from Proposition 8 that ρ′h≤0, so that Hf≥ρ′/ρh. Therefore,
(30)Δfσh=nρhρ′ρh+HfΘ≤nρhρ′ρh−Hf≤0.The proof then follows as in the caseHf>0.Moreover, if warping functionρh satisfies condition (ii), then using (13), we have the next result which extends Theorem 9.Theorem 10.
LetM¯n+1=I×ρMfn be a weighted Riemannian warped product whose fiber M has f-parabolic universal covering. Let ψ:Σn⟶M¯n+1 be a complete two-sided hypersurface that lies in a slab. Suppose the warping function ρh satisfies conditions (ii). If the f-mean curvature Hf satisfies Hf2≤1/Θ2ρ′h2/ρh2, then Σn is a slice.Proof.
From (13), we have
(31)1ρhΔfρh=nρ′h2ρh2+logρ′′h∇h2+nρ′hρhHfΘ≥nρ′h2ρh2+logρ′′h∇h2−n2ρ′h2ρh2+Hf2Θ2≥n2ρ′h2ρh2−Hf2Θ2+logρ′′h∇h2.By the hypothesis, we haveΔfρh≥0. Moreover, since Σn lies in a slab, and ρh is a positive smooth function, then there exists a positive constant C such that ρh≤C. So, we can apply Proposition 7 to get ρh as constant. Therefore, Σn is a slice.Now, we consider then+1-dimensional weighted hyperbolic space ℍn+1, which instead of the more commonly used weighted half-space model, as the weighted warped product ℝ×etℝfn. It can be easily seen that the slices t0×ℝfn of ℍn+1=ℝ×etℝfn are precisely the horospheres. Furthermore, according to Theorem 10, we have the following application in weighted hyperbolic space.Corollary 11.
Letℍn+1=ℝ×etℝfn be a weighted hyperbolic space whose fiber ℝn has f-parabolic universal covering and let ψ:Σn⟶ℍn+1 be a complete two-sided hypersurface which is contained in a slab. If Hf2≤1/Θ2, then Σn 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.
LetM¯n+1=I×ρMfn be a weighted Riemannian warped product whose fiber M has f-parabolic universal covering. Let ψ:Σn⟶M¯n+1 be a complete two-sided hypersurface which lies in a slab. Suppose the warping function ρh satisfies condition (ii), and there is a point h0∈I such that ρ′h0=0. If Hf does not change sign, then Hf=0 and Σn is a slice.Proof.
Since the hypersurfaceΣn is contained in a slab, then h is bounded, and supΣh=h∗,infΣh=h∗. Reasoning as in the proof of Theorem 9, we have the weak maximum principle for f-Laplace operator Δf holds on Σn; then, there exist two sequences pj,qj⊂Σn such that
(32)limjhpj=h∗,limjΔfhpj≤0,limjhqj=h∗,limjΔfhqj≥0.From (12), we have that
(33)limjΔfhpj=limjρ′ρhpjn−∇hpj2+nΘpjHfpj≤0,(34)limjΔfhqj=limjρ′ρhqjn−∇hqj2+nΘqjHfqj≥0.Take−1≤Θ<0. Thus, it follows from (33) and (34) that
(35)Hfpj≥−ρ′h∗/ρh∗n−∣∇hpj2nΘpj,and
(36)Hfqj≤−ρ′h∗/ρh∗n−∇hqj2nΘqj.Furthermore, taking into account thatρ′/ρ is nondecreasing yields
(37)ρ′ρh∗≤ρ′ρh0≤ρ′ρh∗.On the other hand, by (6), we have n−∇h2>0. Therefore, using (35) and (36), we conclude that
(38)Hfpj≥−ρ′h0/ρh0n−∇hpj2nΘpj=0,Hfqj≤−ρ′h0/ρh0n−∇hqj2nΘqj=0.Considering thatHf does not change the sign on Σn, hence Hf=0, that is, Σn is a f-minimal hypersurface. Using (13), we have
(39)Δfρh=nρ′h2ρh+∇h2ρhlogρ′′h≥0.In the following, by the same argument as in Theorem10, we have Σn is a slice.


## 4. Uniqueness Results in Weighted Product Spaces

In this section, we establish some uniqueness results concerning the complete hypersurfacesΣn in weighted product spaces I×Mfn. Firstly, as a consequence of Theorem 9, when the ambient space is weighted product space, we have the following result.Corollary 13.
LetM¯n+1=I×Mfn be a weighted product space whose fiber M has f-parabolic universal covering, and let ψ:Σn⟶M¯n+1 be a complete hypersurface with nonvanishing f-mean curvature which is contained in a slab. If Hf does not change the sign on Σn, then Σn is a slice.Proof.
From (12), we have
(40)Δfh=nHfΘ.SinceHf does not change sign, we have Δfh does not change the sign on Σn, and proceeding as in the proof of Theorem 9, we obtain that Σn is parabolic. Furthermore, we know that height function h is bounded, which implies that Σn is a slice from Proposition 7.Theorem 14.
Letψ:Σn⟶I×Mfn be a complete hypersurface with nonvanishing constant f-mean curvature in a weighted product space I×Mfn whose fiber M has f-parabolic universal covering. Assume that KM≥k for some nonnegative constant k and the weighted function f is convex. If −1≤Θ≤−2/2, then Σn is a slice.Proof.
LetE1,⋯,En be a (local) orthonormal frame in XΣ; using the Gauss equation, we have
(41)RicX,X=∑i=1nR¯X,EiX,Ei+nHAX,X−AX2,
for anyX∈XΣ. Moreover, we also have
(42)R¯X,EiX,Ei=KMX∗,Ei∗X∗,X∗Ei∗,Ei∗−X∗,Ei∗2,
whereKM is the sectional curvature of M, X∗=X−X,∂t∂t and Ei∗=Ei−Ei,∂t∂t are the projections of the tangent vector fields X and Ei onto M, respectively.Considering the hypothesis thatKM≥k for some nonnegative constant k, and by direct computation, we have
(43)∑i=1nR¯X,EiX,Ei≥kn−1X2−n−2X,∇h2−X2∇h2.Summing up,(44)RicX,X≥kn−1X2−n−2X,∇h2−X2∇h2+nHAX,X−AX2.Furthermore, taking into account that the weighted functionf is convex, we have
(45)HessfX,X=Hess¯fX,X+∇¯f,NAX,X≥∇¯f,NAX,X.So(46)RicfX,X≥kn−1X2−n−2X,∇h2−X2∇h2+nHfAX,X−AX2.Particularly,(47)Ricf∇h,∇h≥kn−11−∇h2∇h2+nHfA∇h,∇h−A∇h2.On the other hand, using the Bochner-Lichnerowiz formula (see [2]),
(48)12Δf∇h2=Hessh2+Ricf∇h,∇h+∇Δfh,∇h,where
(49)HesshX=∇X∇h=ΘAX,foranyX∈XΣ.Thus, from (5), we have that
(50)Hessh2=A2Θ2=A21−∇h2.UsingHf as a constant and (12), we have
(51)∇Δh=nHf∇Θ=−nHfA∇h.Consequently,(52)12Δf∇h2≥kn−11−∇h2∇h2+A21−2∇h2.Next, by the hypothesis−1≤Θ≤−2/2 and (6), we have ∇h2≤1/2. So, from (52), it follows that
(53)Δf∇h2≥2kn−11−∇h2∇h2≥0.Moreover, reasoning as in the proof of Theorem9, we have that Σn is parabolic. Since ∇h2 is bounded on Σn, thus by Proposition 7, we conclude that ∇h2 is constant. So, Δf∇h2=0. From (53), we also have ∇h2=0 on Σn; consequently, h is constant, that is, Σn is a slice.To prove our next result, we need the following auxiliary lemma.Lemma 15 ([3]).
Letψ:Σn⟶I×Mfn be a hypersurface with constant f-mean curvature immersed in a weighted product space I×Mfn; then
(54)ΔfΘ=−A2+Ric~fN∗,N∗Θ,
whereRic~f is the Bakry-Émery-Ricci tensor of the fiber M and N∗=N−N,∂t∂t is the orthonormal projection of N onto M.Theorem 16.
Letψ:Σn⟶I×Mfn be a complete hypersurface with nonvanishing constant f-mean curvature in a weighted product space I×Mfn whose fiber M has f-parabolic universal covering. Assume that KM≥−k for some positive constant k and the weighted function f is convex. If ∇h2≤αA2/n−1k, for some constant 0<α<1, then Σn is a slice.Proof.
By the Gauss equation and with a direct computation, we have(55)Ric~fN∗,N∗≥−kn−1∇h2.In the case where−1≤Θ<0, from Lemma 15 and the condition ∇h2≤αA2/n−1k, we have
(56)ΔfΘ≥−A2−kn−1∇h2Θ≥−1α−1kn−1∇h2Θ≥0.On the other hand, as we did before in the proof of Theorem9, it follows that Σn is parabolic. Moreover, since Θ is bounded on Σn, we conclude from Proposition 7 that Θ is constant. So, ΔfΘ=0. Using (56), we have ∇h2=0, which implies that h is constant. Therefore, Σn is a slice.As a consequence of the proof of Theorem16, we can get the following corollary.Corollary 17.
Letψ:Σn⟶I×Mfn be a complete hypersurface with nonvanishing constant f-mean curvature in a weighted product space I×Mfn whose fiber M has f-parabolic universal covering. Assume that KM≥−k and Hess¯f≥−β for some positive constants k and β. If ∇h2≤αA2/n−1k+β, for some constant 0<α<1, then Σn is a slice.


## 5. Nonparametric Results for the Entire Graphs

In this section, we consider the vertical graphs in a weighted Riemannian warped productM¯n+1=I×ρMfn, which are defined by
(57)Σnu=ux,x:x∈Ω⊂M¯n+1,where Ω⊆M be a connected domain of M and u is a smooth function on Ω. Moreover, the metric induced on Ω from the metric on ambient space M¯n+1 via Σnu, which is represented by
(58),=du2+ρ2u,M.It is easy to see from the metric induced onΩ of Σnu that if the function ρu is bounded on Ω, the graphs Σnu is complete. Furthermore, the graph Σnu is said to be entire if Ω=M.In the following, we can give the reason as in the proof of the Theorem9 to obtain a nonparametric result.Corollary 18.
LetM¯n+1=I×ρMfn be a weighted Riemannian warped product whose fiber M has f-parabolic universal covering, and let Σnu be an entire graph with nonvanishing f-mean curvature which lies in a slab. Suppose the warping function ρh satisfies condition i or ii. If Hf2≤ρ′2/ρ2h, then Σnu is a slice.Next, it is not difficult to obtain the following nonparametric version of Theorem10.Corollary 19.
LetM¯n+1=I×ρMfn be a weighted Riemannian warped product whose fiber M has f-parabolic universal covering. Let Σnu be an entire graph which lies in a slab. Suppose the warping function ρh satisfies condition (ii). If the f-mean curvature Hf satisfies Hf2≤1/Θ2ρ′h2/ρh2, then Σnu is a slice.If, moreover,ρ=1, we can have the following corollaries of all other theorems of Section 4.Corollary 20.
LetΣnu be an entire graph with nonvanishing constant f-mean curvature in a weighted product space I×Mfn whose fiber M has f-parabolic universal covering. Assume that KM≥k for some nonnegative constant k and the weighted function f is convex. If −1≤Θ≤−2/2, then Σnu is a slice.Corollary 21.
LetΣnu be an entire graph with nonvanishing constant f-mean curvature in a weighted product space I×Mfn whose fiber M has f-parabolic universal covering. Assume that KM≥−k for some positive constant k and the weighted function f is convex. If ∇h2≤αA2/n−1k, for some constant 0<α<1, then Σnu is a slice.



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