Conformal Geometry of Hypersurfaces in Lorentz Space Forms

Let x : Mn → Mn+1 1 (c) be a space-like hypersurface without umbilical points in the Lorentz space form Mn+1 1 (c). We define the conformal metric and the conformal second fundamental form on the hypersurface, which determines the hypersurface up to conformal transformation ofM 1 (c). We calculate the Euler-Lagrange equations of the volume functional of the hypersurface with respect to the conformal metric, whose critical point is called aWillmore hypersurface, and we give a conformal characteristic of the hypersurfaces with constant mean curvature and constant scalar curvature. Finally, we prove that if the hypersurface x with constant mean curvature and constant scalar curvature is Willmore, then x is a hypersurface inH 1 (−1).


Introduction
Let  :   →  + be an immersed submanifold in sphere  + .In [1], on the submanifold the Wang has constructed a complete invariant system of the Möbius transformation group of  + .Especially for the hypersurface, the Möbius invariants, the Möbius metric, and the Möbius second fundamental form determine the hypersurface up to Möbius transformations provided the dimension of hypersurface  ≥ 3 (also see [2]).After that, the study of the Möbius geometry has been a topic of increasing interest (see [3][4][5][6]).
In this paper we study space-like hypersurfaces in the Lorentz space form  +1 1 () under the conformal transformation group.We follow Wang's idea and construct conformal invariants of space-like hypersurfaces which determine hypersurfaces up to a conformal transformation.
For the Lorentz space form, there exists a united conformal compactification Q +1  1 , which is the projectivized light cone in R +2 induced from  +3 2 (see [7,8]).Using conformal compactification Q +1  1 , we define the conformal metric  and the conformal second fundamental form on a hypersurface in the Lorentz space form, which determines a hypersurface up to a conformal transformation.Clearly, the volume functional with respect to the conformal metric is a conformal invariant.We call a critical hypersurface of the volume functional Willmore hypersurface.There are many studies about the Willmore hypersurface in the Lorentz space form (see, [7,9,10]).
Our main goal is to calculate the Euler-Lagrange equation for the volume functional by conformal invariants and to find some special Willmore hypersurfaces.We find that maximal hypersurfaces in Lorentz space form are not Willmore in general if the dimension  ≥ 3. We give a conformal characteristic of the hypersurfaces with constant mean curvature and constant scalar curvature.By the conformal characteristic, we prove that if the hypersurfaces are Willmore, then the hypersurfaces must be in  +1 1 (−1).Thus, isoparametric hypersurfaces in  +1 1 (1) and  +1 1 are not Willmore.
We organize the paper as follows.In Section 2, we define the conformal invariants and give conformal congruent theorem of hypersurfaces in the Lorentz space form.In Section 3, we calculate the Euler Lagrange equation for the volume functional.In Section 4, we give a conformal characteristic of space-like hypersurfaces with constant mean curvature and constant scalar curvature.In Section 5, we give some examples of the Willmore hypersurface and prove that some special hypersurfaces are not Willmore in general.
Proof.First, we prove that  is well defined.Suppose that  :  →  +1 , ỹ : Ũ →  +2 are different lifts defined in open subsets  and Ũ of .For the local positive definite metrics ⟨ , ⟩  = ⟨, ⟩, we denote by Δ the Laplace operator, by ∇ the gradient of a function  and by  the normalized scalar curvatures with respect to ⟨ , ⟩  , respectively.Analogously for ⟨ ỹ,  ỹ⟩, we denote by Δ the Laplace operator and by κ the normalized scalar curvatures.On  ∩ Ũ, we have ỹ =   , where  ∈  ∞ ( ∩ Ũ).Therefore, ⟨ ỹ,  ỹ⟩ =  2 ⟨, ⟩.By some computations, we have Using these formula, it follows that Next, we prove that  is invariant under conformal transformations of  +1 1 ().Let  be a conformal transformation of  +1 1 (), and we denote  = (); then, there is a  ∈ (+ 3, 2) acting on Q +1 1 , and defined in open subsets ; then, the submanifold  = () must have a local lift like  =   .Since  preserves the Lorentz inner product and the dilatation of the local lift  will not impact the term (⟨Δ, Δ⟩ −  2 )⟨, ⟩, therefore the 2-form  is invariant under conformal transformations.Now, let  +1 1 () =  +1 1 (1) and take local lift  = (1, ); then where  and  are the second fundamental form and the mean curvature of , respectively.Thus,  is positive definite at any nonumbilical point of   ; analogously for hypersurfaces in  +1 1 and  +1 1 .Thus; we complete the proof of Theorem 2. Geometry 3 Now, we assume that space-like hypersurface  :   →  +1 1 () is umbilical-free; thus, the 2-form  is a positive definite.We call  the conformal metric of hypersurface .There exists a unique lift: such that  = ⟨, ⟩.We call  the conformal position vector of .Theorem 2 implies the following.
Let { 1 , . . .,   } be a local orthonormal basis of   with respect to  with dual basis { 1 , . . .,   }.Denote   =   ().We define where Δ is the Laplace operator of ; then, we have We may decompose  +3 2 such that where V ⊥ span{, ,  1 , . . .,   }.We call V the conformal normal bundle of , which is linear bundle.Let  be a local section of V and ⟨, ⟩ = −1; then, {, ,  1 , . . .,   , } forms a moving frame in  +3 2 along   .We may write the structure equations as follows: where {  , ,   ,   } are 1-forms on   with   = −  .It is clear that  := ∑    ⊗   ,  := ∑    ⊗   ,  are globally defined conformal invariants.We call  the conformal second fundamental form,  the conformal 2tensor, and  conformal 1-form, respectively.If we write then we can define the covariant derivatives of these tensors and curvature tensor with respect to conformal metric : By exterior differentiation of structure equations ( 14) and the definition of the covariant derivative of conformal invariants, we can get the integrable conditions of the structure equations: Since  = ⟨, ⟩, we get From structure equation, we have Furthermore, we have where  is the normalized scalar curvature of .From (24), we see that when  ≥ 3, all coefficients in the structure equations are determined by the conformal metric  and the conformal second fundamental form ; thus, we get the following conformal congruent theorem.
Next, we give the relations between the conformal invariants and isometric invariants of  :   →  +1 1 ().First, we consider space-like hypersurface in  +1 1 .Let  :   →  +1 1 be a space-like hypersurface without umbilical points.Let { 1 , . . .,   } be an orthonormal local basis for the induced metric  = ⟨, ⟩ with dual basis { 1 , . . .,   }.Let  +1 be a normal vector field of , and ⟨ +1 ,  +1 ⟩ = −1.Then, we have the first and second fundamental forms ,  and the mean curvature .We may write  = ∑    ⊗   ;  = ∑  ℎ    ⊗   ;  = (1/) ∑  ℎ  .Denote by Δ  the Laplacian and   the normalized scalar curvature for .By structure equation and Gauss equation of  we get that For  :   →  +1 1 , there is a lift: Therefore, the conformal metric and conformal position vector of  are as follows: Let   =  −   ; then {  | 1 ≤  ≤ } are the local orthonormal basis for , and with the dual basis   =     .Let By some calculations, we can obtain that where   =   () and By a direct calculation, we get the following expression of the conformal invariants , , and : where  , is the Hessian of  for  and   =   ().

The First Variation of the Conformal Volume Functional
Let  0 :   →  +1 1 () be a compact space-like hypersurface with boundary   .We define the generalized Willmore functional (  ) (as the volume functional of the conformal metric ): A critical hypersurface of the conformal volume functional is called a Willmore hypersurface.Let  :   ×  →  +1 1 () be an admissible variation of  0 such that for each small .For each ,   has the conformal metric   .As in Section 2, we have a moving frame {, ,   , } in  +3 2 and the Willmore functional (  ).Let  be a local basis for Geometry 5 the conformal normal bundle V  of   .Denote by  and   the differential operators on   ×  and   , respectively.Then, we have By (31), we can find functions , V  , V : Since {, ,   , } is a moving frame along   × , it follows from (37) and ( 38) that where Ω  = −Ω  , Ω  =   + V  ,  = V.By exterior differentiation of (39), we get Since  * (  × ) =  *   ⊕  * , we have the following decomposition: where {  , ,   ,   } are local functions on  × R. Using (40) and comparing the terms in  *  ∧ , we get where {V , } is the covariant derivative of ∑ V    with respect to   .Here, we have used the notations of conformal invariants {  ,   ,   } for   .By the same way, we get from (40) that where { , } are covariant derivatives of ∑      .Using (42) and (43), we get Therefore, we have Now, we calculate the first variation of the following conformal volume functional: where  is the volume for   .From (42) and (46) we get From the fact that the variation is admissible, we know that V  = 0, V = 0 and   (V) = 0 on   .It follows from (48) and Green's formula that Thus, we have the following theorem.
then  is conformal equivalent to the space-like hypersurface in  +1 1 (−1) with constant mean curvature and constant scalar curvature, and Proof.Let  :   →  +1 1 () be a Willmore space-like hypersurface without umbilical points.Let { 1 , . . .,   } be the local orthonormal basis for  such that and corresponding principal curvatures are ( Since (1),  = 0, (2),  =  + , so  is conformal equivalent to a space-like hypersurface with constant mean curvature and constant scalar curvature.We can assume that mean curvature and scalar curvature of  are constant.Thus, from the Gauss equation of , we get that If |ℎ| 2 −  2 = , then ℎ , = 0; thus, the principal curvatures of  are constant.It is well known that space-like isoparametric hypersurfaces in  +1 1 (−1) are either totally umbilical hypersurfaces or   ( 1 ) ×  − ( 2 ) (see [11]).
Thus, we complete the proof of Theorem 14.