A stability result for translating space-like graphs in Lorentz manifolds
aa r X i v : . [ m a t h . DG ] J a n A stability result for translating space-like graphs inLorentz manifolds
Ya Gao, Jing Mao † , Chuanxi Wu Faculty of Mathematics and Statistics,Key Laboratory of Applied Mathematics of Hubei Province,Hubei University, Wuhan 430062, ChinaEmail: [email protected]
Abstract
In this paper, we investigate space-like graphs defined over a domain Ω ⊂ M n in the Lorentzmanifold M n × R with the metric − ds + σ , where M n is a complete Riemannian n -manifoldwith the metric σ , Ω has piecewise smooth boundary, and R denotes the Euclidean 1-space.We can prove an interesting stability result for translating space-like graphs in M n × R under aconformal transformation. Recent years, the study of submanifolds of constant curvature in product manifolds attracts manygeometers’ attention. For instance, Hopf in 1955 discovered that the complexification of the trace-less part of the second fundamental form of an immersed surface U , with CMC H , in R is aholomorphic quadratic differential Q on U , and then he used this observation to get his well-known conclusion that any immersed CMC sphere S ֒ → R is a standard distance sphere withradius 1 / H . By introducing a generalized quadratic differential e Q for immersed surfaces U inproduct spaces S × R and H × R , with S , H the 2-dimensional sphere and hyperbolic surfacerespectively, Abresch and Rosenberg [1] can extend Hopf’s result to CMC spheres in these targetspaces. Meeks and Rosenberg [12] successfully classified stable properly embedded orientableminimal surfaces in the product space N × R , where N is a closed orientable Riemannian surface.In fact, they proved that such a surface must be a product of a stable embedded geodesic on N with R , a minimal graph over a region of N bounded by stable geodesics, N × { t } for some t ∈ R , or isin a moduli space of periodic multigraphs parameterized by P × R + , where P is the set of primitive(non-multiple) homology classes in H ( N ) . Mazet, Rodr´ıguez and Rosenberg [11] analyzed prop-erties of periodic minimal or CMC surfaces in the product manifold H × R , and they also constructexamples of periodic minimal surfaces in H × R . In [13], Rosenberg, Schulze and Spruck showed Corresponding authorMSC 2020: 53C20, 53C42.Key Words: Mean curvature flow, space-like graphs, translating space-like graphs, maximal space-like graphs, con-stant mean curvature, Lorentz manifolds.
1. Gao, J. Mao,C.-X. Wu 2that a properly immersed minimal hypersurface in N × R + equals some slice N × { c } when N isa complete, recurrent n -dimensional Riemannian manifold with bounded curvature. Very recently,Gao, Mao and Song [9] proved the existence and uniqueness of solutions to the CMC equationwith nonzero Neumann boundary data in product manifold N n × R , where N n is an n -dimensional( n ≥
2) complete Riemannian manifold with nonnegative Ricci curvature. Equivalently, this con-clusion gives the existence of CMC graphic hypersurfaces defined over a compact strictly convexdomain Ω ⊂ N n and having nonvanishing contact angle. Of course, for more information, readerscan check references therein of these papers. Hence, it is interesting and important to considersubmanifolds of constant curvature in the product manifold of type N n × R .Inspired by Shahriyari’s progress on complete translating graphs in R (see [16] for details)and the Jenkins-Serrin theory on minimal graphs and CMC graphs, Zhou [18] considered completetranslating, minimal and CMC graphs in 3-dimensional product manifold N × R over a domain Ω ⊂ N , where N is a complete Riemannian surface, and successfully showed the boundary be-havior of Ω . This conclusion extends some of Shahriyari’s conclusions in [16] from the Euclidean3-space R to the setting of 3-dimensional product space N × R . Stability plays an important role in the study of minimal or CMC hypersurfaces in Euclideanspace or, more generally, product manifolds. For instance, if stability assumption was made, nicecurvature estimates or classification results for minimal or CMC surfaces can be obtained – see,e.g., [7, 8, 12, 15, 17, 18].The famous Bernstein theorem (holds only for n ≤
7) in the Euclidean space says that theentire nonparametric minimal hypersurfaces in R n + , n ≤
7, are hyperplanes (see [14]). Calabi[4] (for n ≤ n ) proved that a complete maximal spacelike hypersurfacein the flat Lorentz-Minkowski ( n + ) -space L n + ≡ R n is totally geodesic. Therefore, specially,the only entire nonparametric maximal space-like hypersurfaces in R n are space-like hyperplanes.This interesting example shows that it is meaningful to ask whether classical results in Riemanniangeometry (or specially the Euclidean space) can be transplanted to pesudo-Riemannian geometry(or specially the pesudo-Euclidean space) or not. This example also shows that, in some aspect,there exists essential difference between the Euclidean space and the pesudo-Euclidean space.Motivated by the previous experience, we try to get stability conclusions in Lorentz manifoldsof type M n × R . Fortunately, so far, we get one – see Theorem 1.1 for details. In order to state ourconclusion clearly, we need to introduce some notions first.Throughout this paper, denote by M n × R , with the metric − ds + σ , an ( n + ) -dimensional( n ≥
2) Lorentz manifold where M n is a complete Riemannian n -manifold with the metric σ . Fora domain Ω ⊂ M n with piecewise smooth boundary, a translating space-like graph in the Lorentz ( n + ) -manifold M n × R is the space-like graph of u ( x ) , where u ( x ) : Ω → R is a solution of thefollowing mean curvature type equationdiv Du p − | Du | ! = c p − | Du | , (1.1)where D is a covariant derivative operator on M n , div ( · ) denotes the divergence operator, and c isa constant. Translating space-like graphs by mean curvature flow (MCF for short) in the Lorentzmanifold M n × R are translating surfaces that can be viewed as a space-like graph of a functionover a domain. In fact, let { x , u ( x ) } be a space-like graphic surface defined over Ω ⊂ M n in theLorentz manifold M n × R , and then, since the mean curvature of the space-like surface is (see [5,. Gao, J. Mao,C.-X. Wu 3Sect. 1] in Section 3 here for this calculation) H = div Du p − | Du | ! , the graph of u is a vertically translating space-like with constant speed c if and only if u is a solutionto the equation (1.1). Recently, Mao and his collaborators [5] showed that along the nonparametricMCF with prescribed contact angle boundary condition in the Lorentz 3-manifold M × R , if M has nonnegative Gaussian curvature, then the evolution of space-like graphs over compact strictlyconvex domains in M exists for all the time and solutions of the flow converge to ones moving onlyby translation. Translating solutions play an important role in the study of type-II singularities ofthe MCF. For instance, Angenent and Vel´azquez [2, 3] gave some examples of convergence whichimplies that type-II singularities of the MCF there are modeled by translating surfaces.Denote by ^ M n × R the ( n + ) -dimensional pseudo-Riemannian manifold { ( x , s ) | x ∈ M n , s ∈ R } equipped with the weighted metric e cs ( − ds + σ i j dx i dx j ) . Clearly, ^ M n × R can be achieved bythe Lorentz ( n + ) -manifold M n × R with a conformal transformation to its Lorentzian metric.Here, we have used Einstein summation convention, that is, summation should be done to repeatedsubscripts and superscripts. In the sequel, without specification, Einstein summation conventionwill be always used. We can prove a stability result for translating space-like graphs as follows: Theorem 1.1.
Assume that u ( x ) is a solution to (1.1). Then Σ = { x , u ( x )) | x ∈ Ω } is a stable,maximal space-like graph in ^ M n × R . The paper is organized as follows. In Section 2, some useful formulas for space-like hypersur-faces in a Lorentz manifold will be recalled. The proof of Theorem 1.1 will be given in Section 3.Meanwhile, as a byproduct, a convergence result related to maximal, CMC or translating space-like graphs in Lorentz manifolds will also be shown. In Section 4, examples related to the existenceof translating space-like graphs in the Lorentz ( n + ) -manifold M n × R will be introduced. Given an ( n + ) -dimensional Lorentz manifold (cid:16) M n + , g (cid:17) , with the metric g , and its space-likehypersurface M n . For any p ∈ M n , one can choose a local Lorentzian orthonormal frame field { e , e , e , . . . , e n } around p such that, restricted to M n , e , e , . . . , e n form orthonormal framestangent to M n . Taking the dual coframe field { w , w , w , . . . , w n } such that the Lorentzian metric g can be written as g = − w + ∑ ni = w i . Making the convention on the range of indices0 ≤ α , β , γ , . . . ≤ n ; 1 ≤ i , j , k . . . ≤ n , and doing differentials to forms w α , one can easily get the following structure equations ( Gauss equation ) R i jkl = R i jkl − ( h ik h jl − h il h jk ) , (2.1) ( Codazzi equation ) h i j , k − h ik , j = R i jk , (2.2) ( Ricci identity ) h i j , kl − h i j , lk = n ∑ m = h m j R mikl + n ∑ m = h im R m jkl , (2.3). Gao, J. Mao,C.-X. Wu 4and the Laplacian of the second fundamental form h i j of M n as follows ∆ h i j = n ∑ k = (cid:0) h kk , i j + R kik , j + R i jk , k (cid:1) + n ∑ k = (cid:0) h kk R i j + h i j R k k (cid:1) + n ∑ m , k = (cid:0) h m j R mkik + h mk R mi jk + h mi R mk jk (cid:1) − n ∑ m , k = (cid:0) h mi h m j h kk + h km h m j h ik − h km h mk h i j − h mi h mk h k j (cid:1) , (2.4)where R and R are the curvature tensors of M n and M n + respectively, A : = h i j w i w j is the secondfundamental form with h i j the coefficient components of the tensor A , ∆ is the Laplacian on thehypersurface M n , and, as usual, the comma “,” in subscript of a given tensor means doing covariantderivatives – this convention will also be used in the sequel. For detailed derivation of the aboveformulae, we refer readers to, e.g., [10, Section 2].Clearly, in our setting here, all formulas mentioned in this section can be used directly with M n + = M n × R . Similar to the calculation in [5, Sect. 1], for the space-like graph Σ = { ( x , u ( x )) | x ∈ Ω } , definedover Ω ⊂ M n , in the Lorentz ( n + ) -manifold M n × R with the metric g : = σ i j dw i ⊗ dw j − ds ⊗ ds ,tangent vectors are given by X i = ∂ i + D i u ∂ s , i = , , . . . , n , and the corresponding upward unit normal vector is given by ~ v = p − | Du | (cid:0) ∂ s + D j u ∂ j (cid:1) , where D j u = σ i j D i u . Denote by ∇ the gradient operator on M n × R , and then the second funda-mental form h i j dw i ⊗ dw j of Σ is given by h i j = −h ∇ X i X j ,~ v i = p − | Du | D i D j u . Moreover, the scalar mean curvature of Σ is H = n ∑ i = h ii = p − | Du | n ∑ i , k = g ik D k D i u ! = ∑ ni , k = (cid:16) σ ik + D i uD k u −| Du | (cid:17) D k D i u p − | Du | = div Du p − | Du | ! , (3.1)where g ik is the inverse of the induced Riemannian metric g on the space-like graph Σ . Denote by Θ the angle function of Σ , and then using (1.1), the above equality can be written equivalently as H = − c Θ = − c h ~ v , ∂ s i . (3.2). Gao, J. Mao,C.-X. Wu 5 Proof of Theorem 1.1.
The area functional of ^ M n × R is given by F ( Σ ) = Z Σ e cs d µ , where d µ is the volume element of Σ induced by the metric g of the Lorentz ( n + ) -manifold M n × R . Let Σ r be a family of surfaces satisfying ∂Σ r ∂ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = = φ ~ v with Σ = Σ , (3.3)where φ ( x ) is a smooth function defined on Σ with compact support. Treating Σ r as a curvatureflow of Σ in the Lorentz ( n + ) -manifold M n × R , and by direct calculation, it follows that: Lemma 3.1.
Along the curvature flow (3.3), we have ∂ ~ v ∂ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = = ∇φ , ∂ H ∂ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = = ∆φ − (cid:0) | A | + Ric ( ~ v ,~ v ) (cid:1) φ , (3.4) where, following the convention used in Section 2, ∇ and ∆ denote the covariant derivative and theLaplacian of Σ respectively, and Ric ( · , · ) stands for the Ricci tensor of the ambient space M n × R .Proof. First, we have ∂ ~ v ∂ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = = (cid:28) ∂ ~ v ∂ r , ( Σ r ) , i (cid:29) g ik ( Σ r ) , k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = = − h ~ v , ( φ ~ v ) , i i g ik ( Σ r ) , k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = = φ , i g ik ( Σ r ) , k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = = ∇φ , where, following the convention used in Section 2, ( · ) , k means doing covariant derivative withrespect to the tangent vector X k on the translating space-like graph Σ .Second, we have ∂ g lm ∂ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = = ∂∂ r (cid:10) ( Σ r ) , l , ( Σ r ) , m (cid:11) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = = (cid:10) ( φ ~ v ) , l , ( Σ r ) , m (cid:11) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = = − φ (cid:10) ~ v , ( Σ r ) , lm (cid:11) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = = φ h i j , . Gao, J. Mao,C.-X. Wu 6and ∂ h i j ∂ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = = − ∂∂ r (cid:10) ~ v , ( Σ r ) , i j (cid:11) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = = − D φ , l ( Σ r ) , m g ml , ( Σ r ) , i j E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = − (cid:10) ~ v , ( φ ~ v ) , i j (cid:11) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = = − D φ , l ( Σ r ) , m g ml , Γ ki j ( Σ r ) , k + h i j ~ v E (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = − (cid:28) ~ v , (cid:16) φ , j ~ v + φ h jl g lm ( Σ r ) , m (cid:17) , i (cid:29) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = = − Γ ki j φ , k + φ , i j + φ h jl g lm h im = ∇ i ∇ j φ + φ h il g lm h im , where, as usual, Γ ki j denote Christoffel symbols determined by the metric g . By (2.1), (2.2), (2.4)and Simon’s identity of φ , we have g i j ∂ h i j ∂ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = = ∆φ + | A | φ − Ric ( ~ v ,~ v ) φ , and then ∂ H ∂ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = = ∂∂ r (cid:0) g i j h i j (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = = − g il ∂ g lm ∂ r | r = g m j h i j + g i j ∂ h i j ∂ r | r = = − φ | A | + ∆φ + | A | φ − Ric ( ~ v ,~ v ) φ = ∆φ − (cid:0) | A | + Ric ( ~ v ,~ v ) (cid:1) φ . This completes the proof of Lemma 3.1.By (3.2) and (3.4), it is not hard to obtain ∂ F ( Σ r ) ∂ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = = Z Σ φ ( H + c h ~ v , ∂ s i ) e cs d µ = , ∂ F ( Σ r ) ∂ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = = Z Σ φ (cid:2) ∆φ − (cid:0) | A | + Ric ( ~ v ,~ v ) (cid:1) φ + c h ∇φ , ∂ s i (cid:3) e cs d µ . (3.5)Define an elliptic operator L as follows L φ = ∆φ − (cid:0) | A | + Ric ( ~ v ,~ v ) (cid:1) φ + c h ∇φ , ∂ s i . (3.6)Therefore, putting (3.6) into the second equality of (3.5) yields ∂ F ( Σ r ) ∂ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = = Z Σ φ L φ e cs d µ . (3.7). Gao, J. Mao,C.-X. Wu 7Now, we only need to show that the RHS of (3.7) is non-positive. Since Σ is a space-like graph, itsangle function satisfies Θ = h ~ v , ∂ r i <
0. Thus we can write φ = ηΘ , where η is another functionover Σ with compact support. Then it follows that φ L φ = ηΘ ( η L Θ + Θ∆η + h ∇η , ∇Θ i + c Θ h ∇η , ∂ s i ) . (3.8)The reason why we adapt this form is based on a general formula of ∆Θ as follows. Lemma 3.2.
For any C space-like hypersurface S in the Lorentz ( n + ) -manifold M n × R , it holdsthat ∆Θ − (cid:0) | A | + Ric ( ~ v ,~ v ) (cid:1) Θ − h ∇ H , ∂ s i = , (3.9) where A is the second fundamental form of S.Proof. Fix a point p ∈ S . Suitably choose an orthonormal frame field { e , e , . . . , e n } on S suchthat ∇ e i e j ( p ) = h e i , e j i = δ i j . Then ∇ e i e j ( p ) = h i j ~ v , where, following the convention used inSection 2, ∇ denotes the covariant derivative of the ambient space M n × R and ~ v is the unit normalvector of S . It is easy to know that for any smooth vector field X , ∇ X ∂ s =
0. By direct calculation,one has ∆Θ ( p ) = ∇ e i ∇ e i h ∂ s ,~ v i − ∇ ∇ ei e i Θ ( p )= e i h ∂ s , h ik e k i ( p )= h ik , i h ∂ s , e k i + | A | Θ . (3.10)Using the Codazzi equation (2.2) directly yields h ik , i = h ii , k + R iki . Hence, it gives h ik , i h ∂ s , e k i = h ∇ H , ∂ s i + Ric ( ~ v , h ∂ s , e k i e k ) . (3.11)Since ∇ X ∂ s = X , we know h ∂ s , e k i e k = ∂ s + Θ ~ v and Ric ( ~ v , ∂ s ) =
0. Putting these two facts into (3.11) implies h ik , i h ∂ s , e k i = h ∇ H , ∂ s i + Ric ( ~ v ,~ v ) Θ . The assertion of this lemma follows by combing the above equality with (3.10) directly.Let us go back to the proof of Theorem 1.1. Since Σ is a translating space-like graph in theLorentz ( n + ) -manifold M n × R , one has H = − c Θ , and then (3.9) can be rewritten as L Θ = . Therefore (3.8) becomes φ L φ = ηΘ ( Θ∆η + h ∇η , ∇Θ i + c Θ h ∇η , ∂ s i ) . . Gao, J. Mao,C.-X. Wu 8On the other hand, the divergence of ηΘ ∇η e cs isdiv (cid:0) ηΘ ∇η e cs (cid:1) = ηΘ e cs ( Θ∆η + h ∇η , ∇Θ i + c Θ h ∇η , ∂ s i ) + Θ | ∇η | e cs = φ e cs L φ + Θ | ∇η | e cs . (3.12)Combining (3.12) with (3.7) and applying the divergence theorem result in ∂ F ( Σ r ) ∂ r (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) r = = − Z Σ Θ | ∇η | e cs d µ ≤ . Then we conclude that the translating space-like graph Σ is stable and maximal in ^ M n × R .Applying Lemma 3.2, we can obtain the following interesting rigidity result. Theorem 3.3.
Let { Σ n } ∞ n = be a sequence of smooth connected space-like graphs in the Lorentz ( n + ) -manifold M n × R with diameter ρ converging uniformly to a connected space-like hyper-surface Σ in the C sense. If all Σ n are translating space-like graphs in the interior of Σ , the anglefunction Θ satisfies that Θ < or Θ ≡ . The conclusion is also true in the case of maximal orCMC space-like graphs.Proof. Without loss of generality, we assume Θ <
0. By continuity, we know that in the interiorof all Σ n , | A | < β holds for some positive constant β depending only on M n .Now, first, we assume that Σ n are maximal or CMC space-like graphs. Then ∇ H ≡
0. ByLemma 3.2, we have ∆Θ − (cid:0) | A | + Ric ( ~ v ,~ v ) (cid:1) Θ = Σ n . SinceRic ( ~ v ,~ v ) = u , k ( Γ ikk , i + Γ lkk Γ iil − Γ iik , k − Γ lik Γ ikl ) − | Du | , i , k , l = , , . . . , n , there exists a positive constant β only depending on M n such that Ric ( ~ v ,~ v ) ≤ β in the interiorof all Σ n . By (3.13) we have ∆Θ ≥ ( β + β ) Θ on all Σ n . Because Σ is the C uniform limit of Σ n as n → ∞ , it follows that Θ ≤ ∆Θ ≥ ( β + β ) Θ . By the strong maximum principle ofsecond-order elliptic equations, we can obtain that Θ ≡ Θ < Σ .Second, assume that Σ n are translating space-like graphs. Then H ≡ − c Θ by (3.2). Similarargument gives ∆Θ ≥ ( β + β ) Θ − c h ∇Θ , ∂ s i on all Σ n . Based on the strong maximum principle and the fact that Θ ≤ Σ , we also have Θ ≡ Θ < Σ .. Gao, J. Mao,C.-X. Wu 9 In this section, we construct some examples of translating space-like graphs to MCF when thehypersurface M n has a domain with certain warped product structure.Suppose that M n is an n -dimensional ( n ≥
2) complete Riemannian manifold with a metric σ containing a domain M n equipped with the following coordinate system: (cid:8) θ = ( θ , θ , . . . , θ n ) ∈ S n − , r ∈ [ , r ) (cid:9) with σ = dr + h ( r ) d θ , (4.1)where d θ is the round metric on the unit ( n − ) -sphere S n − , h ( r ) is a positive function satisfying h ( ) = h ′ ( ) = h ′ ( r ) = r ∈ ( , r ) .Now, with the help of examples constructed below, we can somehow show the existence oftranslating space-like graphs in the Lorentz ( n + ) -manifold M n × R with the structure (4.1) andthe metric g . Theorem 4.1.
Let M n be a complete Riemannian n-manifold mentioned above. Let u ( r ) : [ , r ) → R be a C solution of the following ordinary differential equation (ODE for short)u rr − u r + ( n − ) h ′ ( r ) h ( r ) u r = c , (4.2) with u r ( ) = for r ∈ [ , r ) and | u r | < . Then Σ = ( x , u ( r )) for r ∈ [ , r ) is a translating space-like graph in the Lorentz ( n + ) -manifold M n × R , where x = ( r , θ ) ∈ M n given by (4.1) . If r = ∞ ,then Σ is complete. Remark 4.2.
Clearly, (4.2) is a second-order ODE whose component of the second-order deriva-tive term does not degenerate under the assumption | u r | <
1. The existence of its solution isobvious.
Proof. If r = ∞ , then M n is simply connected and should be a whole M n . Thus Σ is complete.In the rest part, we show that Σ is a translating space-like graph. By (3.2), we know that here itis sufficient to derive the identity H = − c Θ , (4.3)where H is the mean curvature of Σ and ~ v is its upward normal vector.Fix a point ( x , u ( x )) on Σ , where x ∈ M n and the polar coordinate of x in M n is not ( , , . . . , ) .Clearly, the polar coordinate system on M n given by (4.1) determines a frame field { ∂ r , ∂ θ , . . . , ∂ θ n } naturally. For the space-like graph Σ determined by u ( x ) = u ( r ) in the Lorentz ( n + ) -manifold M n × R , denote by u r and u θ i , i = , , . . . , n , the partial derivatives of u . Since here u ( r ) is aradial function, u θ i ≡ i = , , . . . , n . Therefore, on Σ , a natural frame { e = ∂ r + u r ∂ s , e i = ∂ θ i } , i = , . . . , n can be obtained, where, as before, ∂ s denotes the vector field tangent to R . Then theRiemannian metric on Σ and the upward unit normal vector of Σ are given by g = h e , e i = − u r , g kl = g lk = h e l , e k i = , k = l , g ii = h e i , e i i = h ( r ) , i = , . . . , n , .Gao, J. Mao,C.-X. Wu 10and ~ v = ∂ s + u r ∂ r p − u r . By direct calculation, its second fundamental forms are h = − D ∇ e e ,~ v E = u rr p − u r , and h ii = − D ∇ e i e i ,~ v E = − (cid:10) − h ( r ) h ′ ( r ) ∂ r ,~ v (cid:11) = h ′ ( r ) h ( r ) u r p − u r , i = , . . . , n . where we use the fact D ∇ e i e i , ∂ r E = − h ′ ( r ) h ( r ) , i = , . . . , n . Then, by (4.2), the mean curvature of Σ with respect to ~ v is H = g h + g h + . . . + g nn h nn = p − u r (cid:18) u rr − u r + ( n − ) h ′ ( r ) h ( r ) u r (cid:19) = c p − u r . On the other hand, we have Θ = h ~ v , ∂ s i = * ∂ s + u r ∂ r p − u r , ∂ s + = − p − u r . Hence in our case here we have H = − c Θ , which implies Σ is a translating space-like graph. Theproof is finished. Acknowledgments
This research was supported in part by the NSF of China (Grant Nos. 11801496 and 11926352), theFok Ying-Tung Education Foundation (China), and Hubei Key Laboratory of Applied Mathematics(Hubei University).
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