# Conformal Geometry of Hypersurfaces in Lorentz Space Forms

Authors: Tongzhu Li; Changxiong Nie
Journal: Geometry
 (2013)

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## Abstract

Letx:Mn→M1n+1(c) be a space-like hypersurface without umbilical points in the Lorentz space form M1n+1(c). We define the conformal metric and the conformal second fundamental form on the hypersurface, which determines the hypersurface up to conformal transformation of M1n+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 a Willmore 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 in H1n+1(-1).


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## Body

## 1. Introduction

Letx:Mn→Sn+p be an immersed submanifold in sphere Sn+k. In [1], on the submanifold the Wang has constructed a complete invariant system of the Möbius transformation group of Sn+p. 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 n≥3 (also see [2]). After that, the study of the Möbius geometry has been a topic of increasing interest (see [3–6]).In this paper we study space-like hypersurfaces in the Lorentz space formM1n+1(c) 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 compactificationℚ1n+1, which is the projectivized light cone in ℝPn+2 induced from R2n+3 (see [7, 8]). Using conformal compactification ℚ1n+1, we define the conformal metric g 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 dimensionn≥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 H1n+1(-1). Thus, isoparametric hypersurfaces in S1n+1(1) and R1n+1 are not Willmore.We organize the paper as follows. In Section2, 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.


## 2. Conformal Invariants of Hypersurfaces in Lorentz Space

In this section, we define some conformal invariants of hypersurface and give a congruent theorem of hypersurfaces under the conformal group ofM1n+1(c).LetRtn+2 be the real vector space Rn+2 with the Lorenz inner product 〈,〉 given by
(1)〈X,Y〉:=-∑i=1t‍xiyi+∑j=t+1n+2‍xjyj.
Let M1n+1(c) be a Lorentz space form. When c=0, M1n+1(c)=R1n+1; when c=1, M1n+1(c)=S1n+1(1),  and when c=-1, M1n+1(c)=H1n+1(-1), where
(2)S1n+1(1)={x∈R1n+2∣〈x,x〉=1},H1n+1(-1)={x∈R2n+2∣〈x,x〉=-1}.
We denote by Cn+2 the cone in ℝ2n+3 and by ℚ1n+1 the conformal compactification space in ℝPn+3:
(3)Cn+2:={X∈ℝ2n+3∣〈X,X〉=0,X≠0},ℚ1n+1:={[X]∈ℝPn+2∣〈X,X〉=0}.
Let O(n+3,2) be the Lorentz group of R2n+3 keeping the Lorentz inner product 〈,〉 invariant. Then, O(n+3,2) is a transformation group on ℚ1n+1 defined by
(4)T([X]):=[XT],X∈Cn+2,T∈O(n+3,2).
Topologically, ℚ1n+1 is identified with the compact space Sn×S1/S0, which is endowed by a standard Lorentz metric h=gSn⊕(-gS1). ℚ1n+1 has conformal metric:
(5)[h]:={eτh∣τ∈C∞(ℚ1n+1)},and[O(n+3,2)] is the conformal transformation group of ℚ1n+1.We define the following mappings (without ambiguity all denote byσ):
(6)σ:R1n+1⟶ℚ1n+1,u↦[(〈u,u〉+12,u,〈u,u〉-12)],σ:S1n+1(1)⟶ℚ1n+1,u↦[(1,u)],σ:H1n+1(-1)⟶ℚ1n+1,u↦[(u,1)].
Using σ, we can regard hypersurfaces in M1n+1(c) as submanifolds in ℚ1n+1. A classical theorem states the following.Theorem 1.
Two hypersurfacesx,x¯:Mn→M1n+1(c) are conformal equivalent if and only if there exists T∈O(n+3,2) such that σ∘x=T(σ∘x¯):Mn→ℚ1n+1.Letx:Mn→M1n+1(c) be a space-like hypersurface without umbilical points; then, (σ∘x)*(TMn) is a positive definite subbundle of Tℚ1n+1. For any local lift Z of the standard projection π:Cn+2→ℚ1n+1, we get a local lift y:=Z∘σ∘x:U→Cn+1 of σ∘x:M→ℚ1n+1, which is defined in an open subset U of Mn. Thus, 〈dy,dy〉=λ2dx·dx is a local metric, which is conformal to the induced metric dx·dx. We denote by Δ and κ the Laplacian operator and the normalized scalar curvature with respect to the local positive definite metric 〈dy,dy〉, respectively. Then, we have the following.Theorem 2.
Letx:Mn→M1n+1(c) be a space-like hypersurface; then the 2-form g:=-(〈Δy,Δy〉-n2κ)〈dy,dy〉 is a globally defined conformal invariant. Moreover, g is positive definite at any nonumbilical point of Mn.Proof.
First, we prove thatg is well defined. Suppose that y:U→Cn+1, y~:U~→Cn+2 are different lifts defined in open subsets U and U~ of M. For the local positive definite metrics 〈,〉y=〈dy,dy〉, we denote by Δ the Laplace operator, by ∇f the gradient of a function f and by κ the normalized scalar curvatures with respect to 〈,〉y, respectively. Analogously for 〈dy~,dy~〉, we denote by Δ~ the Laplace operator and by κ~ the normalized scalar curvatures. On U∩U~, we have y~=eτy, where τ∈C∞(U∩U~). Therefore, 〈dy~,dy~〉=e2τ〈dy,dy〉. By some computations, we have
(7)e2τΔ~f=Δf+(n-2)〈∇τ,∇f〉y,e2τκ~=κ-2mΔτ-n-2n〈∇τ,∇τ〉y.
Using these formula, it follows that(8)(〈Δy,Δy〉-n2κ)〈dy,dy〉=(〈Δ~y~,Δ~y~〉-n2κ~)〈dy~,dy~〉.
Next, we prove thatg is invariant under conformal transformations of M1n+1(c). Let ϕ be a conformal transformation of M1n+1(c), and we denote x¯=ϕ(x); then, there is a T∈O(n+3,2) acting on ℚ1n+1, and y:U→Cn+2 is a lift of σ∘x:M→ℚ1n+1 defined in open subsets U; then, the submanifold x¯=ϕ(x) must have a local lift like y¯=eτyT. Since T preserves the Lorentz inner product and the dilatation of the local lift y will not impact the term (〈Δy,Δy〉-m2κ)〈dy,dy〉, therefore the 2-form g is invariant under conformal transformations.
Now, letM1n+1(c)=S1n+1(1) and take local lift y=(1,x); then
(9)-(〈Δy,Δy〉-n2κ)=nn-1(|II|2-nH2),
whereII and H are the second fundamental form and the mean curvature of x, respectively. Thus, g is positive definite at any nonumbilical point of Mn; analogously for hypersurfaces in R1n+1 and H1n+1. Thus; we complete the proof of Theorem 2.Now, we assume that space-like hypersurfacex:Mn→M1n+1(c) is umbilical-free; thus, the 2-form g is a positive definite. We call g the conformal metric of hypersurface x. There exists a unique lift:
(10)Y:M⟶Cn+2such thatg=〈dY,dY〉. We call Y the conformal position vector of x. Theorem 2 implies the following.Theorem 3.
Two space-like hypersurfacesx,x¯:Mn→M1n+1(c) are conformal equivalent if and only if there exists T∈O(n+3,2) such that Y¯=YT, where Y,Y~ are the conformal position vectors of x,x¯, respectively.Let{E1,…,En} be a local orthonormal basis of Mn with respect to g with dual basis {ω1,…,ωn}. Denote Yi=Ei(Y). We define
(11)N:=-1nΔY-12n2〈ΔY,ΔY〉Y,whereΔ is the Laplace operator of g; then, we have
(12)〈N,Y〉=1,〈N,N〉=0,〈N,Yk〉=0,〈Yi,Yj〉=δij,1≤k≤n.We may decomposeR2n+3 such that
(13)R2n+3=span{Y,N}⊕span{Y1,…,Yn}⊕𝕍,where𝕍⊥span{Y,N,Y1,…,Yn}. We call 𝕍 the conformal normal bundle of x, which is linear bundle. Let ξ be a local section of 𝕍 and 〈ξ,ξ〉=-1; then, {Y,N,Y1,…,Yn,ξ} forms a moving frame in R2n+3 along Mn. We may write the structure equations as follows:
(14)dY=∑i‍ωiYi,dN=∑i‍AiYi+Cξ,dYi=-AiY-ωiN+∑j‍ωijYj+Biξ,dξ=CY+∑i‍BiYi,where{Ai,C,Bi,ωij} are 1-forms on Mn with ωij=-ωij. It is clear that A:=∑i‍Ai⊗ωi, B:=∑i‍Bi⊗ωi, C are globally defined conformal invariants. We call B the conformal second fundamental form, A the conformal 2-tensor, and C conformal 1-form, respectively. If we write
(15)Ai=∑j‍Aijωj,Bi=∑j‍Bijωj,C=∑i‍Ciωi,then we can define the covariant derivatives of these tensors and curvature tensor with respect to conformal metricg:
(16)∑j‍Ci,jωj=dCi+∑k‍Ckωkj,∑k‍Aij,kωk=dAij+∑k‍Aikωkj+∑k‍Akjωki,∑k‍Bij,kωk=dBij+∑k‍Bikωkj+∑k‍Bkjωki.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:(17)Aij=Aji,Bij=Bji,(18)Aij,k-Aik,j=BijCk-BikCj,(19)Bij,k-Bik,j=δijCk-δikCj,(20)Ci,j-Cj,i=∑k‍(BikAkj-BjkAki),(21)Rijkl=BilBjk-BikBjl+Aikδjl+Ajlδik-Ailδjk-Ajkδil.Sinceg=〈dY,dY〉, we get
(22)〈ΔY,ΔY〉=n2κ-1.From structure equation, we have(23)ΔY=-tr(A)-nN+tr(B)ξ.Furthermore, we have(24)tr(A)=12n(n2κ-1),Rij=tr(A)δij+(n-2)Aij+∑k‍BikBkj,(1-n)Ci=∑j‍Bij,j,∑ij‍Bij2=n-1n,∑i‍Bii=0,whereκ is the normalized scalar curvature of g. From (24), we see that when n≥3, all coefficients in the structure equations are determined by the conformal metric g and the conformal second fundamental form B; thus, we get the following conformal congruent theorem.Theorem 4.
Two space-like hypersurfacesx:Mn→M1n+1(c) and x¯:Mn→M1n+1(c)(n≥3) are conformal equivalent if and only if there exists a diffeomorphism φ:Mn→Mn which preserves the conformal metric and the conformal second fundamental form.Remark 5.
By using the same method as in [1], we can define conformal invariants of space-like submanifold x:Mm→M1n+1(c). But for time-like submanifolds, the situation is very different. In fact, for space-like submanifolds, the globally defined conformal invariant g:=-(〈Δy,Δy〉-m2κ)〈dy,dy〉 is positive definite or semipositive definite, and for time-like submanifolds, the globally defined conformal invariant g:=-(〈Δy,Δy〉-m2κ)〈dy,dy〉 is non-definite.Next, we give the relations between the conformal invariants and isometric invariants ofx:Mn→M1n+1(c).First, we consider space-like hypersurface inR1n+1. Let x:Mn→R1n+1 be a space-like hypersurface without umbilical points. Let {e1,…,en} be an orthonormal local basis for the induced metric I=〈dx,dx〉 with dual basis {θ1,…,θn}. Let en+1 be a normal vector field of x, and 〈en+1,en+1〉=-1. Then, we have the first and second fundamental forms I,II and the mean curvature H. We may write I=∑i‍θi⊗θi; II=∑ij‍hijθi⊗θj; H=(1/n)∑i‍hii. Denote by ΔM the Laplacian and κM the normalized scalar curvature for I. By structure equation and Gauss equation of x we get that
(25)ΔMx=nHen+1,κM=-1n(n-1)(n2|H|2-|II|2).
For x:Mn→R1n+1, there is a lift:
(26)y:Mn⟶Cn+2,y=(〈x,x〉+12,x,〈x,x〉-12).
So, we get
(27)〈dy,dy〉=〈dx,dx〉=I,Δ=ΔM,κ=κM.
It follows from (25) that
(28)〈ΔY,ΔY〉-n2κ=nn-1(-|II|2+n|H|2).
Therefore, the conformal metric and conformal position vector of x are as follows:
(29)g=nn-1(|II|2-n|H|2)〈dx,dx〉≔e2τI,Y=nn-1(|II|2-n|H|2)(〈x,x〉+12,x,〈x,x〉-12).
Let Ei=e-τei; then {Ei∣1≤i≤n} are the local orthonormal basis for g, and with the dual basis ωi=eτθi. Let
(30)yi=(〈x,ei〉,ei,〈x,ei〉),yn+1=(〈x,en+1〉,en+1,〈x,en+1〉).
By some calculations, we can obtain that
(31)Y=eτy,Yi=eτ(τiy+yi),ξ=-Hy+yn+1,-eτN=12(|∇τ|2-|H|2)y+∑i‍τiyi+Hyn+1+(1,0→,1),whereτi=ei(τ) and |∇τ|2=∑i‍τi2.By a direct calculation, we get the following expression of the conformal invariantsA, B, and C:
(32)Aij=e-2τ[τiτj-hijH-τi,j+12(-|∇τ|2+|H|2)δij],Bij=e-τ(hij-Hδij),Ci=e-2τ(Hτi-Hi-∑j‍hijτj),whereτi,j is the Hessian of τ for I and Hi=ei(H).Using the same methods, we can obtain relations between the conformal invariants and isometric invariants ofx:Mn→S1n+1(1) and x:Mn→H1n+1(-1). We have the following unitied expression of the conformal invariants A, B, and C:
(33)Aij=e-2τ[τiτj-τi,j-hijH+12(-|∇τ|2+|H|2+ϵ)δij],Bij=e-τ(hij-Hδij),Ci=e-2τ(Hτi-Hi-∑j‍hijτj),
where ϵ=1 for x:Mn→S1n+1(1) and ϵ=-1 for x:Mn→H1n+1(-1).


## 3. The First Variation of the Conformal Volume Functional

Letx0:Mn→M1n+1(c) be a compact space-like hypersurface with boundary ∂Mn. We define the generalized Willmore functional W(Mn) (as the volume functional of the conformal metric g):
(34)Volg(Mn)=∫MdMg=(nn-1)n/2∫Mn(|II|2-nH2)n/2dMI.A critical hypersurface of the conformal volume functional is called a Willmore hypersurface.Letx:Mn×R→M1n+1(c) be an admissible variation of x0 such that
(35)x(·,t)|∂Mn=x0|∂Mn,(36)dxt(TpMn)|∂Mn=dx0(TpMn)|∂Mn,
for each small t. For each t, xt has the conformal metric gt. As in Section 2, we have a moving frame {Y,N,Yi,ξ} in R2n+3 and the Willmore functional W(xt). Let ξ be a local basis for the conformal normal bundle 𝕍t of xt. Denote by d and dM the differential operators on Mn×R and Mn, respectively. Then, we have
(37)d=dM+dt∧∂∂t.
By (31), we can find functions w,vi,v:Mn×R→R such that
(38)∂Y∂t=wY+∑i‍viYi+vξ.
Since {Y,N,Yi,ξ} is a moving frame along Mn×R, it follows from (37) and (38) that
(39)dY=wdtY+∑i‍ΩiYi+Vξ,dN=-wdtN+∑i‍ΨiYi+Φξ,dYi=-ΨiY-ΩiN+∑j‍ΩijYj+Fiξ,dξ=-ΦY-VN-∑i‍FiYi,
where Ωij=-Ωji, Ωi=ωi+vidt, V=vdt. By exterior differentiation of (39), we get
(40)dΩi=∑j‍Ωij∧Ωj+wdt∧Ωi+vdt∧Fi,dv∧dt=∑i‍Ωi∧Fi,dFi=∑j‍Fj∧Ωji-Ωi∧Φ-Ψi∧V,dΩij=∑k‍Ωik∧Ωkj-Ωi∧Ψj-Ψi∧Ωj+Fi∧Fj.
Since T*(Mn×R)=T*Mn⊕T*R, we have the following decomposition:
(41)Ωij=ωij+Lijdt,Ψi=Ai+uidt,Φ=C+udt,Fi=Bi+bidt,
where {ui,u,Lij,bi} are local functions on M×ℝ. Using (40) and comparing the terms in T*M∧dt, we get
(42)∂ωi∂t=∑j‍(vi,j+Lij+Bijv)ωj+wωi,
where {vi,j} is the covariant derivative of ∑‍viEi with respect to gt. Here, we have used the notations of conformal invariants {Aij,Bij,Ci} for xt. By the same way, we get from (40) that
(43)bi=ei(v)+∑j‍Bijvj,∂Fi∂t=∑j‍(bi,j+∑k‍LikBkj+Aijv-viCj)ωj+uωi,
where {bi,j} are covariant derivatives of ∑i‍biωi. Using (42) and (43), we get
(44)∂Bij∂t+wBij=bi,j+∑k‍(LikBik-BikLkj)-viCj+Aijv+uδij-∑k‍Bik(vk,j+Bkjv),bi,j=v,ij+∑k‍(Bik,jvk+Bikvk,j),v,ij=dMei(v)+∑j‍ej(v)ωji.
Therefore, we have
(45)∂Bij∂t+wBij=v,ij+∑k‍(LikBik-BikLkj+Bij,kvk)+(Aij-∑k‍BikBkj)+(u-∑k‍vkCk)δij,(46)n-1nw=∑ij‍Bij[v,ij+(Aij-∑k‍BikBkj)v].
Now, we calculate the first variation of the following conformal volume functional:
(47)W(t)=vol(gt)=∫Mn‍ω1∧⋯∧ωn=∫Mn‍dMt,
where dM is the volume for gt. From (42) and (46) we get
(48)W′(t)=∑i‍∫Mn‍ω1∧⋯∧∂ωi∂t∧⋯∧ωn=n∫Mn‍wdMt=n2n-1∫Mn‍{∑ij‍Bij[v,ij+(Aij-∑k‍BikBkj)v]}dMt.
From the fact that the variation is admissible, we know that vi=0, v=0 and ei(v)=0 on ∂Mn. It follows from (48) and Green’s formula that
(49)W′(t)=n2n-1×∫Mn‍{∑ij‍Bij,ij+∑ij‍AijBij-∑ijk‍BikBkjBij}×vdMt.
Thus, we have the following theorem.Theorem 6.
A space-like hypersurfacex:Mn→M1n+1(c) is a Willmore hypersurface (i.e., a critical hypersurface to the conformal volume functional) if and only if
(50)∑ij‍Bij,ij+∑ij‍AijBij-∑ijk‍BikBkjBij=0.Using (24), we can write the Euler-Lagrange equations as
(51)∑i‍Ci,i+∑ij‍(1n-1Rij-Aij)Bij=0.Theorem 7.
Any maximal (or zero mean curvature) space-like surface in Lorentz space forms is a Willmore surface.Proof.
Letx:M2→R13 be a space-like surface. Let {e1,e2} be a local orthonormal basis of 〈dx,dx〉 and e3 a local normal vector field. From (32), we get that
(52)∑ij‍AijBij=∑ij‍e-2τBij(τiτj-τi,j)+∑ij‍e-τBijBijH.
Since we have the following relations of connections:
(53)ωij=θij+τiωj-τjωi,
a direct calculation implies that
(54)∑i‍Ci,i=∑ij‍e-2τBij(τiτj-τi,j)-e-3τΔMH.
From (51), we have
(55)∑i‍Ci,i-∑ij‍AijBij=-e-3τ(ΔMH+e2τ∑ij‍BijBijH).
Ifx is a maximal space-like surface, that is, H=0 and Rij=κδij, thus x is Willmore.
Similarly, we can verify that maximal space-like surfaces inS13(1) and H13(-1) are also Willmore. Thus, we complete the proof of Theorem 7.


## 4. A Characteristic of CMC Hypersurfaces andκM=Constant

In this section, we consider space-like hypersurfaces with constant mean curvature and constant scalar curvature.Proposition 8.
Letx:Mn→M1n+1(c) be a space-like hypersurface without umbilical points. If the mean curvature and scalar curvature of x are constant, then conformal invariants of x satisfy
(56)(1),C=0,(2),A=μB+λg,
where μ, λ are constant.Proof.
First, we consider the space-like hypersurfacex:Mn→R1n+1. Since H and κ are constant, then by the Gauss equation we have that
(57)e2τ=nn-1(|II|2-nH2)=constant.
From (32), we get that
(58)Ci=0,Bij=e-τ(hij-Hδij),Aij=e-2τ[12H2-Hhij].
From (58), Let μ=-e-τH and λ=-(1/2)e-2τH2; then, we prove the formula (56).
Similarly, we can prove the formula (56) for M1n+1(c)=S1n+1(1) and M1n+1(c)=H1n+1(-1). Thus, we complete the proof of Proposition 8.Theorem 9.
Letx:Mn→M1n+1(c) be a space-like hypersurface without umbilical points. If conformal invariants of x satisfy
(59)(1),C=0,(2),A=μB+λg.
Then x is conformal equivalent to a space-like hypersurface with constant mean curvature and constant scalar curvature.Proof.
SinceC=0, from (20), we can take the local orthonormal basis {E1,…,En} such that
(60)(Bij)=diag(b1,…,bn),(Aij)=diag(a1,…,an).
Since A=μB+λg, from structure equations (14) we get that
(61)dN-λdY-μdξ=0.
Taking exterior differentiation of (61), we get that
(62)dλ∧dY+dμ∧dξ=0.
Writing dλ=∑i‍λiωi, dμ=∑i‍μiωi, from (62) we get that
(63)∑ijk‍(λkδij+μkBij)ωk∧ωjYi=0,
which implies that
(64)λkδij+μkBij=λjδik+μjBik.
Set i=j in (64) and taking summation over i, we get that
(65)λk-1n-1μkbk=0.
Set i=j≠k in (65), from (60), we get that
(66)λk+μkbi=0,i≠k.
Combining (65) and (66), we get that
(67)μk(bi+1n-1bk)=0,i≠k.
Ifμi=0, 1≤i≤n, then λi=0, 1≤i≤n.
Ifμi≠0, for some i; we can assume that μ1≠0. Then, from (67), we have
(68)bi=-1n-1b1,i≠1.
Since tr(B)=0 and |B|2=(n-1)/n, thus we have
(69)b1=n-1n,b2=⋯=bn=-1n.
Using dBij+∑k‍Bkjωki+∑k‍Bikωkj=∑k‍Bij,kωk and (69), we get that
(70)ω1i=0,1<i≤n.
Thus, we have
(71)R1i1i=0,1<i≤n,a2=⋯=an.
Since ai=μbi+λ, then from (71), we have
(72)2λ+n-2nμ=n-1n2,(73)2λ1+n-2nμ1=0.
From (66), we have
(74)λ1+1nμ1=0.
From (73) and (74), we get that
(75)μ1=0.
This is a contradiction; soμi=0, 1≤i≤n. Since μi=0, λi=0, 1≤i≤n, so λ and μ are constant.
From (61), we have
(76)d(N-λY-μξ)=0.
Therefore, we can find a constant vector e∈R2n+3 such that
(77)N-λY-μξ=e.
From (11) and (77), we get that
(78)〈e,e〉=μ2-2λ,〈Y,e〉=1.
To prove the theorem, we consider the following three cases.
Case  1. e is light-like, that is, μ2-2λ=0.
Case  2. e is space-like, that is, μ2-2λ>0.
Case  3. e is time-like, that is, μ2-2λ<0.
First, we consider Case  1;e is ligh-tlike, that is, μ2-2λ=0. Then, there exists a T∈O(n+2,2) such that
(79)e¯=(-1,0→,1)=eT=(N-λY-μξ)T.
Let x¯:Mn→R1n+1 be a hypersurface that its conformal position vector is Y¯=YT, then N¯=NT, ξ¯=ξT, and
(80)e¯=N¯-λY¯-μξ¯.
Writing
(81)Y¯=eτ¯(〈x¯,x¯〉+12,x¯,〈x¯,x¯〉-12),ξ¯=-H¯(〈x¯,x¯〉+12,x¯,〈x¯,x¯〉-12)+y¯n+1,
then from
(82)〈Y¯,e¯〉=1,〈ξ¯,e¯〉=-μ,
we obtain that
(83)eτ¯=1,H¯=μ.
Since Y¯=((〈x¯,x¯〉+1)/2,x¯,(〈x¯,x¯〉-1)/2), then g=〈dx¯,dx¯〉=I¯. From (24) we have
(84)tr(A)=nλ=12n(n2κ-1).
SinceκM=κ, so the mean curvature and scalar curvature of hypersurface x¯ are constant.
Next, we consider Case  2;e is space-like, that is, μ2-2λ>0. Then, there exists a T∈O(n+2,2) such that
(85)e¯=(0→,μ2-2λ)=eT=(N-λY-μξ)T.
Let x¯:Mn→H1n+1(-1) be a hypersurface which its conformal position vector is Y¯=YT; then N¯=NT, ξ¯=ξT, and
(86)e¯=N¯-λY¯-μξ¯.
Writing Y¯=eτ¯(x¯,1), ξ¯=-H¯(x¯,1)+y¯n+1, then from
(87)〈Y¯,e¯〉=1,〈ξ¯,e¯〉=-μ,
we obtain that
(88)eτ¯=1μ2-2λ,H¯=μ.
Since 〈dx¯,dx¯〉=(μ2-2λ)g,
(89)κM=1μ2-2λκ.
Therefore, the mean curvature and scalar curvature of hypersurfacex¯ are constant.
Finally, we consider Case  3;e is time-like, that is, μ2-2λ<0. Then, there exists a T∈O(n+2,2) such that
(90)e¯=(-2λ-μ2,0→)=eT=(N-λY-μξ)T.
Let x¯:Mn→S1n+1(1) be a hypersurface that its conformal position vector is Y¯=YT; then N¯=NT, ξ¯=ξT, and
(91)e¯=N¯-λY¯-μξ¯.
Writing Y¯=eτ¯(1,x¯), ξ¯=-H¯(1,x¯)+y¯n+1, then from
(92)〈Y¯,e¯〉=1,〈ξ¯,e¯〉=-μ,
we obtain that
(93)eτ¯=12λ-μ2,H¯=μ.
Since 〈dx¯,dx¯〉=(2λ-μ2)g,
(94)κM=12λ-μ2κ.
Therefore, the mean curvature and scalar curvature of hypersurfacex¯ are constant. Thus, we complete the proof the Theorem 9.Corollary 10.
Letx:Mn→M1n+1(c) be a space-like hypersurface without umbilical points. If conformal invariants of x satisfy
(95)(1),C=0,(2),A=λg,
then x is conformal equivalent to a maximal space-like hypersurface with constant scalar curvature in H1n+1(-1).Proof.
From the process of proof of Theorem9, we have that x is conformal equivalent to a maximal space-like hypersurface with constant scalar curvature. Next, we prove that x is a maximal space-like hypersurface with constant scalar curvature in H1n+1(-1). In fact, using covariant derivative of ∇B and C=0, we have
(96)12Δ|B|2=∑ijk‍|Bij,k|2+∑ij‍ΔBij=∑ijk‍|Bij,k|2+(n-1n)2+(n-1)ntr(A)+n∑ijk‍BikBkjAji.
Since |B|2=(n-1)/n and A=λg, we get that
(97)∑ijk‍|Bij,k|2+(n-1n)2+2(n-1)λ=0.
Thus, λ≤-(n-1)/2n2<0. From the process of proof of the Theorem 9, we have that x is conformal equivalent to a maximal space-like hypersurface with constant scalar curvature in H1n+1(-1).


## 5. Some Special Willmore Space-Like Hypersurfaces

In this section, we consider some special Willmore hypersurfaces. First, we give an example of the Willmore space-like hypersurface with constant mean curvature and constant scalar curvature inH1n+1(-1).Example 11.
Let(98)Hk(r1)={u=(u0,u→1)∈R1k+1∣〈u,u〉=-r12},Hn-k(r2)={v=(v0,v→1)∈R1n-k+1∣〈v,v〉=-r22},
where r12+r22=1. Define space-like hypersurface
(99)x:Hk(r1)×Hn-k(r2)⟶H1n+1(-1),x(u,v)=(u0,v0,u→1,v→1)∈H1n+1(-1)⊂R2n+2.
Then, unit normal vector of x is en+1=((r2/r1)u0,-(r1/r2)v0,(r2/r1)u→1,-(r1/r2)v→1); thus, the induced metric and the second fundamental form of x are, respectively,
(100)I=Iu+Iv,II=r2r1Iu-r1r2Iv,nH=kr2r1-(n-k)r1r2,|II|2=k(r2r1)2+(n-k)(r1r2)2.
Now, we assume that x is Willmore. Since H and |II|2 are constant, from (33), we get that the Euler-Lagrange equation (51) is
(101)k(r2r1)6+(3k-2n)(r2r1)4+(3k-n)(r2r1)2+k=0.
Since r12+r22=1, we get that
(102)r1=-n-kn,r2=-kn.Corollary 12.
Space-like hypersurfacex:Hk(r1)×Hn-k(r2)→H1n+1(-1) is Willmore if and only if
(103)r1=-n-kn,r2=-kn.Remark 13.
Space-like hypersurfacex:Hk(r1)×Hn-k(r2)→H1n+1(-1) is maximal if and only if
(104)r1=-kn,r2=-n-kn.
Thus, maximal hypersurfaces are not Willmore in general.Theorem 14.
Letx:Mn→M1n+1(c) be a Willmore space-like hypersurface without umbilical points. If conformal invariants of x satisfy
(105)(1),C=0,(2),A=μB+λg,
then x is conformal equivalent to the space-like hypersurface in H1n+1(-1) with constant mean curvature and constant scalar curvature, and
(106)2|h|2H=tr(h3)+nH3,|h|2-nH2≤n,
where h=(hij). If |h|2-nH2=n, then x is
(107)Hk(-n-kn)×Hn-k(-kn).Proof.
Letx:Mn→M1n+1(c) be a Willmore space-like hypersurface without umbilical points. Let {E1,…,En} be the local orthonormal basis for g such that
(108)(Bij)=diag(b1,…,bn),
and corresponding principal curvatures are (k1,…,k2). Since
(109)(1),C=0,(2),A=μB+λg,
and since x is Willmore, from (51), we get that
(110)n-1nμ=∑i‍bi3.
From (33), we have
(111)2|h|2H=tr(h3)+nH3.
By calculation of (1/2)Δ|h|2, we have
(112)12Δ|h|2=∑ijk‍|hij,k|2-n2|∇H|2+12∑ij‍Rijij(ki-kj)2=∑ijk‍|hij,k|2-n2|∇H|2+nc|h|2-n2cH2+|h|4-nH∑i‍ki3.
Since (1),C=0, (2),A=μB+λg, so x 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 x are constant. Thus, from the Gauss equation of x, we get that
(113)|h|2=constant,H=constant.
From (112), we have
(114)0=∑ijk‍|hij,k|2+nc|h|2-n2cH2+|h|4-nH∑i‍ki3.
Combining (111) and (114), we get that
(115)∑ijk‍|hij,k|2+nc(|h|2-nH2)+(|h|2-nH2)2=0.
Since |h|2-nH2≥0 and from (115), we get that
(116)c=-1.
Thus, x is a space-like hypersurface in H1n+1(-1). From (115), we have
(117)∑ijk‍|hij,k|2+(|h|2-nH2)(|h|2-nH2-n)=0.
Since |h|2-nH2≥0, from (117), we have
(118)|h|2-nH2≤n.
If|h|2-nH2=n, then hij,k=0; thus, the principal curvatures of x are constant. It is well known that space-like isoparametric hypersurfaces in H1n+1(-1) are either totally umbilical hypersurfaces or Hk(r1)×Hn-k(r2) (see [11]).
Thus, we complete the proof of Theorem14.



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*Source: 549602-2013-09-16.xml*