Conformal Geometry of Hypersurfaces in Lorentz Space Forms

Abstract

Let 𝑥∶𝑀^𝑛→𝑀₁^𝑛+1(𝑐) be a space-like hypersurface without umbilical points in the Lorentz space form 𝑀₁^𝑛+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 𝑀₁^𝑛+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 𝐻₁^𝑛+1(−1).

1. Introduction

Let 𝑥∶𝑀^𝑛→𝑆^𝑛+𝑝 be an immersed submanifold in sphere 𝑆^𝑛+𝑘. In [1], on the submanifold the Wang has constructed a complete invariant system of the Möbius transformation group of 𝑆^𝑛+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 form 𝑀₁^𝑛+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 compactification ℚ₁^𝑛+1, which is the projectivized light cone in ℝ𝑃^𝑛+2 induced from 𝑅₂^𝑛+3 (see [7, 8]). Using conformal compactification ℚ₁^𝑛+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 dimension 𝑛≥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 𝐻₁^𝑛+1(−1). Thus, isoparametric hypersurfaces in 𝑆₁^𝑛+1(1) and 𝑅₁^𝑛+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.

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 of 𝑀₁^𝑛+1(𝑐).

Let 𝑅_𝑡^𝑛+2 be the real vector space 𝑅^𝑛+2 with the Lorenz inner product ⟨,⟩ given by ⟨𝑋,𝑌⟩∶=−^𝑡_𝑖=1𝑥_𝑖𝑦_𝑖+^𝑛+2_𝑗=𝑡+1𝑥_𝑗𝑦_𝑗. (1) Let 𝑀₁^𝑛+1(𝑐) be a Lorentz space form. When 𝑐=0, 𝑀₁^𝑛+1(𝑐)=𝑅₁^𝑛+1; when 𝑐=1, 𝑀₁^𝑛+1(𝑐)=𝑆₁^𝑛+1(1), and when 𝑐=−1, 𝑀₁^𝑛+1(𝑐)=𝐻₁^𝑛+1(−1), where 𝑆₁^𝑛+1(1)=𝑥∈𝑅₁^𝑛+2∣⟨⟩,𝐻𝑥,𝑥=1_1𝑛+1(−1)=𝑥∈𝑅₂^𝑛+2∣⟨𝑥,𝑥⟩=−1. (2) We denote by 𝐶^𝑛+2 the cone in ℝ₂^𝑛+3 and by ℚ₁^𝑛+1 the conformal compactification space in ℝ𝑃^𝑛+3: 𝐶^𝑛+2∶=𝑋∈ℝ₂^𝑛+3∣⟨⟩,ℚ𝑋,𝑋=0,𝑋≠0_1𝑛+1∶=[𝑋]∈ℝ𝑃^𝑛+2∣⟨𝑋,𝑋⟩=0. (3) Let 𝑂(𝑛+3,2) be the Lorentz group of 𝑅₂^𝑛+3 keeping the Lorentz inner product ⟨,⟩ invariant. Then, 𝑂(𝑛+3,2) is a transformation group on ℚ₁^𝑛+1 defined by 𝑇([𝑋])∶=[𝑋𝑇],𝑋∈𝐶^𝑛+2,𝑇∈𝑂(𝑛+3,2). (4) Topologically, ℚ₁^𝑛+1 is identified with the compact space 𝑆^𝑛×𝑆¹/𝑆⁰, which is endowed by a standard Lorentz metric ℎ=𝑔_𝑆𝑛⊕(−𝑔_𝑆1). ℚ₁^𝑛+1 has conformal metric: [ℎ]∶=𝑒^𝜏ℎ∣𝜏∈𝐶^∞ℚ₁^𝑛+1, (5)

and [𝑂(𝑛+3,2)] is the conformal transformation group of ℚ₁^𝑛+1.

We define the following mappings (without ambiguity all denote by 𝜎): 𝜎∶𝑅₁^𝑛+1⟶ℚ₁^𝑛+1⟨⟩,𝑢↦𝑢,𝑢+12⟨⟩,𝑢,𝑢,𝑢−12,𝜎∶𝑆₁^𝑛+1(1)⟶ℚ₁^𝑛+1,𝑢↦[(1,𝑢)],𝜎∶𝐻₁^𝑛+1(−1)⟶ℚ₁^𝑛+1,𝑢↦[(𝑢,1)]. (6) Using 𝜎, we can regard hypersurfaces in 𝑀₁^𝑛+1(𝑐) as submanifolds in ℚ₁^𝑛+1. A classical theorem states the following.

Theorem 1. Two hypersurfaces 𝑥,𝑥∶𝑀^𝑛→𝑀₁^𝑛+1(𝑐) are conformal equivalent if and only if there exists 𝑇∈𝑂(𝑛+3,2) such that 𝜎∘𝑥=𝑇(𝜎∘𝑥)∶𝑀^𝑛→ℚ₁^𝑛+1.

Let 𝑥∶𝑀^𝑛→𝑀₁^𝑛+1(𝑐) be a space-like hypersurface without umbilical points; then, (𝜎∘𝑥)∗(𝑇𝑀^𝑛) is a positive definite subbundle of 𝑇ℚ₁^𝑛+1. For any local lift 𝑍 of the standard projection 𝜋∶𝐶^𝑛+2→ℚ₁^𝑛+1, we get a local lift 𝑦∶=𝑍∘𝜎∘𝑥∶𝑈→𝐶^𝑛+1 of 𝜎∘𝑥∶𝑀→ℚ₁^𝑛+1, which is defined in an open subset 𝑈 of 𝑀^𝑛. 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. Let 𝑥∶𝑀^𝑛→𝑀₁^𝑛+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 𝑀^𝑛.

Proof. First, we prove that 𝑔 is well defined. Suppose that 𝑦∶𝑈→𝐶^𝑛+1, 𝑦∶𝑈→𝐶^𝑛+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=𝑒^𝜏𝑦, where 𝜏∈𝐶^∞(𝑈∩𝑈). Therefore, ⟨𝑑𝑦,𝑑𝑦⟩=𝑒^2𝜏⟨𝑑𝑦,𝑑𝑦⟩. By some computations, we have 𝑒^2𝜏Δ𝑓=Δ𝑓+(𝑛−2)∇𝜏,∇𝑓_𝑦,𝑒^2𝜏2𝜅=𝜅−𝑚Δ𝜏−𝑛−2𝑛⟨∇𝜏,∇𝜏⟩_𝑦. (7)
Using these formula, it follows that Δ𝑦,Δ𝑦−𝑛²𝜅𝑑𝑦,𝑑𝑦=Δ𝑦,Δ𝑦−𝑛²𝜅𝑑𝑦,𝑑𝑦. (8)
Next, we prove that 𝑔 is invariant under conformal transformations of 𝑀₁^𝑛+1(𝑐). Let 𝜙 be a conformal transformation of 𝑀₁^𝑛+1(𝑐), and we denote 𝑥=𝜙(𝑥); then, there is a 𝑇∈𝑂(𝑛+3,2) acting on ℚ₁^𝑛+1, and 𝑦∶𝑈→𝐶^𝑛+2 is a lift of 𝜎∘𝑥∶𝑀→ℚ₁^𝑛+1 defined in open subsets 𝑈; then, the submanifold 𝑥=𝜙(𝑥) must have a local lift like 𝑦=𝑒^𝜏𝑦𝑇. 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, let 𝑀₁^𝑛+1(𝑐)=𝑆₁^𝑛+1(1) and take local lift 𝑦=(1,𝑥); then −Δ𝑦,Δ𝑦−𝑛²𝜅=𝑛||𝑛−1𝐼𝐼²−𝑛𝐻², (9)
where 𝐼𝐼 and 𝐻 are the second fundamental form and the mean curvature of 𝑥, respectively. Thus, 𝑔 is positive definite at any nonumbilical point of 𝑀^𝑛; analogously for hypersurfaces in 𝑅₁^𝑛+1 and 𝐻₁^𝑛+1. Thus; we complete the proof of Theorem 2.

Now, we assume that space-like hypersurface 𝑥∶𝑀^𝑛→𝑀₁^𝑛+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: 𝑌∶𝑀⟶𝐶^𝑛+2 (10)

such that 𝑔=⟨𝑑𝑌,𝑑𝑌⟩. We call 𝑌 the conformal position vector of 𝑥. Theorem 2 implies the following.

Theorem 3. Two space-like hypersurfaces 𝑥,𝑥∶𝑀^𝑛→𝑀₁^𝑛+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,…,𝐸_𝑛} be a local orthonormal basis of 𝑀^𝑛 with respect to 𝑔 with dual basis {𝜔₁,…,𝜔_𝑛}. Denote 𝑌_𝑖=𝐸_𝑖(𝑌). We define 1𝑁∶=−𝑛1Δ𝑌−2𝑛²⟨Δ𝑌,Δ𝑌⟩𝑌, (11)

where Δ is the Laplace operator of 𝑔; then, we have ⟨𝑁,𝑌⟩=1,⟨𝑁,𝑁⟩=0,𝑁,𝑌_𝑘𝑌=0,_𝑖,𝑌_𝑗=𝛿_𝑖𝑗,1≤𝑘≤𝑛. (12)

We may decompose 𝑅₂^𝑛+3 such that 𝑅₂^𝑛+3=span{𝑌,𝑁}⊕span𝑌₁,…,𝑌_𝑛⊕𝕍, (13)

where 𝕍⟂span{𝑌,𝑁,𝑌₁,…,𝑌_𝑛}. 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,…,𝑌_𝑛,𝜉} forms a moving frame in 𝑅₂^𝑛+3 along 𝑀^𝑛. We may write the structure equations as follows: 𝑑𝑌=_𝑖𝜔_𝑖𝑌_𝑖,𝑑𝑁=_𝑖𝐴_𝑖𝑌_𝑖+𝐶𝜉,𝑑𝑌_𝑖=−𝐴_𝑖𝑌−𝜔_𝑖𝑁+_𝑗𝜔_𝑖𝑗𝑌_𝑗+𝐵_𝑖𝜉,𝑑𝜉=𝐶𝑌+_𝑖𝐵_𝑖𝑌_𝑖, (14)

where {𝐴_𝑖,𝐶,𝐵_𝑖,𝜔_𝑖𝑗} are 1-forms on 𝑀^𝑛 with 𝜔_𝑖𝑗=−𝜔_𝑖𝑗. It is clear that 𝐴∶=∑_𝑖𝐴_𝑖⊗𝜔_𝑖, 𝐵∶=∑_𝑖𝐵_𝑖⊗𝜔_𝑖, 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 𝐴_𝑖=_𝑗𝐴_𝑖𝑗𝜔_𝑗,𝐵_𝑖=_𝑗𝐵_𝑖𝑗𝜔_j,𝐶=_𝑖𝐶_𝑖𝜔_𝑖, (15)

then we can define the covariant derivatives of these tensors and curvature tensor with respect to conformal metric 𝑔: _𝑗𝐶_𝑖,𝑗𝜔_𝑗=𝑑𝐶_𝑖+_𝑘𝐶_𝑘𝜔_𝑘𝑗,_𝑘𝐴_{𝑖𝑗,𝑘}𝜔_𝑘=𝑑𝐴_𝑖𝑗+_𝑘𝐴_𝑖𝑘𝜔_𝑘𝑗+_𝑘𝐴_𝑘𝑗𝜔_𝑘𝑖,_𝑘𝐵_{𝑖𝑗,𝑘}𝜔_𝑘=𝑑𝐵_𝑖𝑗+_𝑘𝐵_𝑖𝑘𝜔_𝑘𝑗+_𝑘𝐵_𝑘𝑗𝜔_𝑘𝑖. (16)

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:𝐴_𝑖𝑗=𝐴_𝑗𝑖,𝐵_𝑖𝑗=𝐵_𝑗𝑖,𝐴_{𝑖𝑗,𝑘}−𝐴_{𝑖𝑘,𝑗}=𝐵_𝑖𝑗𝐶_𝑘−𝐵_𝑖𝑘𝐶_𝑗,𝐵_{𝑖𝑗,𝑘}−𝐵_{𝑖𝑘,𝑗}=𝛿_𝑖𝑗𝐶_𝑘−𝛿_𝑖𝑘𝐶_𝑗,𝐶_𝑖,𝑗−𝐶_𝑗,𝑖=_𝑘𝐵_𝑖𝑘𝐴_𝑘𝑗−𝐵_𝑗𝑘𝐴_𝑘𝑖,𝑅_{𝑖𝑗𝑘𝑙}=𝐵_𝑖𝑙𝐵_𝑗𝑘−𝐵_𝑖𝑘𝐵_𝑗𝑙+𝐴_𝑖𝑘𝛿_𝑗𝑙+𝐴_𝑗𝑙𝛿_𝑖𝑘−𝐴_𝑖𝑙𝛿_𝑗𝑘−𝐴_𝑗𝑘𝛿_𝑖𝑙. (17)(18)(19)(20)(21)

Since 𝑔=⟨𝑑𝑌,𝑑𝑌⟩, we get ⟨Δ𝑌,Δ𝑌⟩=𝑛²𝜅−1. (22)

From structure equation, we have Δ𝑌=−tr(𝐴)−𝑛𝑁+tr(𝐵)𝜉. (23)

Furthermore, we have (𝐴)=1tr𝑛2𝑛²,𝑅𝜅−1_𝑖𝑗=tr(𝐴)𝛿_𝑖𝑗+(𝑛−2)𝐴_𝑖𝑗+_𝑘𝐵_𝑖𝑘𝐵_𝑘𝑗,(1−𝑛)𝐶_𝑖=_𝑗𝐵_{𝑖𝑗,𝑗},_𝑖𝑗𝐵²_𝑖𝑗=𝑛−1𝑛,_𝑖𝐵_𝑖𝑖=0, (24)

where 𝜅 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 hypersurfaces 𝑥∶𝑀^𝑛→𝑀₁^𝑛+1(𝑐) and 𝑥∶𝑀^𝑛→𝑀₁^𝑛+1(𝑐)(𝑛≥3) are conformal equivalent if and only if there exists a diffeomorphism 𝜑∶𝑀^𝑛→𝑀^𝑛 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∶𝑀^𝑚→𝑀₁^𝑛+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 of 𝑥∶𝑀^𝑛→𝑀₁^𝑛+1(𝑐).

First, we consider space-like hypersurface in 𝑅₁^𝑛+1. Let 𝑥∶𝑀^𝑛→𝑅₁^𝑛+1 be a space-like hypersurface without umbilical points. Let {𝑒₁,…,𝑒_𝑛} be an orthonormal local basis for the induced metric 𝐼=⟨𝑑𝑥,𝑑𝑥⟩ with dual basis {𝜃₁,…,𝜃_𝑛}. Let 𝑒_𝑛+1 be a normal vector field of 𝑥, and ⟨𝑒_𝑛+1,𝑒_𝑛+1⟩=−1. Then, we have the first and second fundamental forms 𝐼,𝐼𝐼 and the mean curvature 𝐻. We may write 𝐼=∑_𝑖𝜃_𝑖⊗𝜃_𝑖; 𝐼𝐼=∑_𝑖𝑗ℎ_𝑖𝑗𝜃_𝑖⊗𝜃_𝑗; 𝐻=(1/𝑛)∑_𝑖ℎ_𝑖𝑖. Denote by Δ_𝑀 the Laplacian and 𝜅_𝑀 the normalized scalar curvature for 𝐼. By structure equation and Gauss equation of 𝑥 we get that Δ_𝑀𝑥=𝑛𝐻𝑒_𝑛+1,𝜅_𝑀=−1𝑛()𝑛𝑛−1²|𝐻|²−|𝐼𝐼|². (25) For 𝑥∶𝑀^𝑛→𝑅₁^𝑛+1, there is a lift: 𝑦∶𝑀^𝑛⟶𝐶^𝑛+2⟨⟩,𝑦=𝑥,𝑥+12⟨⟩,𝑥,𝑥,𝑥−12. (26) So, we get 𝑑𝑦,𝑑𝑦=⟨𝑑𝑥,𝑑𝑥⟩=𝐼,Δ=Δ_𝑀,𝜅=𝜅_𝑀. (27) It follows from (25) that ⟨Δ𝑌,Δ𝑌⟩−𝑛²𝑛𝜅=−||𝑛−1𝐼𝐼²+𝑛|𝐻|². (28) Therefore, the conformal metric and conformal position vector of 𝑥 are as follows: 𝑛𝑔=||𝑛−1𝐼𝐼²−𝑛|𝐻|²⟨𝑑𝑥,𝑑𝑥⟩∶=𝑒^2𝜏𝐼,𝑌=𝑛||𝑛−1𝐼𝐼²−𝑛|𝐻|²⟨𝑥,𝑥⟩+12⟨⟩,𝑥,𝑥,𝑥−12. (29) Let 𝐸_𝑖=𝑒^−𝜏𝑒_𝑖; then {𝐸_𝑖∣1≤𝑖≤𝑛} are the local orthonormal basis for 𝑔, and with the dual basis 𝜔_𝑖=𝑒^𝜏𝜃_𝑖. Let 𝑦_𝑖=𝑥,𝑒_𝑖,𝑒_𝑖,𝑥,𝑒_𝑖,𝑦_𝑛+1=𝑥,𝑒_𝑛+1,𝑒_𝑛+1,𝑥,𝑒_𝑛+1. (30) By some calculations, we can obtain that 𝑌=𝑒^𝜏𝑦,𝑌_𝑖=𝑒^𝜏𝜏_𝑖𝑦+𝑦_𝑖,𝜉=−𝐻𝑦+𝑦_𝑛+1,−𝑒^𝜏1𝑁=2|∇𝜏|²−|𝐻|²𝑦+_𝑖𝜏_𝑖𝑦_𝑖+𝐻𝑦_𝑛+1+1,0,1, (31)

where 𝜏_𝑖=𝑒_𝑖(𝜏) and |∇𝜏|²=∑_𝑖𝜏²_𝑖.

By a direct calculation, we get the following expression of the conformal invariants 𝐴, 𝐵, and 𝐶: 𝐴_𝑖𝑗=𝑒^−2𝜏𝜏_𝑖𝜏_𝑗−ℎ_𝑖𝑗𝐻−𝜏_𝑖,𝑗+12−|∇𝜏|²+|𝐻|²𝛿_𝑖𝑗,𝐵_𝑖𝑗=𝑒^−𝜏ℎ_𝑖𝑗−𝐻𝛿_𝑖𝑗,𝐶_𝑖=𝑒^−2𝜏𝐻𝜏_𝑖−𝐻_𝑖−_𝑗ℎ_𝑖𝑗𝜏_𝑗, (32)

where 𝜏_𝑖,𝑗 is the Hessian of 𝜏 for 𝐼 and 𝐻_𝑖=𝑒_𝑖(𝐻).

Using the same methods, we can obtain relations between the conformal invariants and isometric invariants of 𝑥∶𝑀^𝑛→𝑆₁^𝑛+1(1) and 𝑥∶𝑀^𝑛→𝐻₁^𝑛+1(−1). We have the following unitied expression of the conformal invariants 𝐴, 𝐵, and 𝐶: 𝐴_𝑖𝑗=𝑒^−2𝜏𝜏_𝑖𝜏_𝑗−𝜏_𝑖,𝑗−ℎ_𝑖𝑗1𝐻+2−|∇𝜏|²+|𝐻|²+𝜖𝛿_𝑖𝑗,𝐵_𝑖𝑗=𝑒^−𝜏ℎ_𝑖𝑗−𝐻𝛿_𝑖𝑗,𝐶_𝑖=𝑒^−2𝜏𝐻𝜏_𝑖−𝐻_𝑖−_𝑗ℎ_𝑖𝑗𝜏_𝑗, (33) where 𝜖=1 for 𝑥∶𝑀^𝑛→𝑆₁^𝑛+1(1) and 𝜖=−1 for 𝑥∶𝑀^𝑛→𝐻₁^𝑛+1(−1).

3. The First Variation of the Conformal Volume Functional

Let 𝑥₀∶𝑀^𝑛→𝑀₁^𝑛+1(𝑐) be a compact space-like hypersurface with boundary 𝜕𝑀^𝑛. We define the generalized Willmore functional 𝑊(𝑀^𝑛) (as the volume functional of the conformal metric 𝑔): Vol_𝑔𝑀^𝑛=∫_𝑀𝑑𝑀_𝑔=𝑛𝑛−1^𝑛/2∫_𝑀𝑛|𝐼𝐼|²−𝑛𝐻^2𝑛/2𝑑𝑀_𝐼. (34)

A critical hypersurface of the conformal volume functional is called a Willmore hypersurface.

Let 𝑥∶𝑀^𝑛×𝑅→𝑀₁^𝑛+1(𝑐) be an admissible variation of 𝑥₀ such that 𝑥(⋅,𝑡)|_𝜕𝑀^𝑛=𝑥_{0𝜕𝑀𝑛},𝑑𝑥_𝑡𝑇_𝑝𝑀^𝑛_𝜕𝑀𝑛=𝑑𝑥₀𝑇_𝑝𝑀^𝑛_𝜕𝑀𝑛, (35)(36) for each small 𝑡. For each 𝑡, 𝑥_𝑡 has the conformal metric 𝑔_𝑡. As in Section 2, we have a moving frame {𝑌,𝑁,𝑌_𝑖,𝜉} in 𝑅₂^𝑛+3 and the Willmore functional 𝑊(𝑥_𝑡). Let 𝜉 be a local basis for the conformal normal bundle 𝕍_𝑡 of 𝑥_𝑡. Denote by 𝑑 and 𝑑_𝑀 the differential operators on 𝑀^𝑛×𝑅 and 𝑀^𝑛, respectively. Then, we have 𝑑=𝑑_𝑀𝜕+𝑑𝑡∧.𝜕𝑡 (37) By (31), we can find functions 𝑤,𝑣_𝑖,𝑣∶𝑀^𝑛×𝑅→𝑅 such that 𝜕𝑌𝜕𝑡=𝑤𝑌+_𝑖𝑣_𝑖𝑌_𝑖+𝑣𝜉. (38) Since {𝑌,𝑁,𝑌_𝑖,𝜉} is a moving frame along 𝑀^𝑛×𝑅, it follows from (37) and (38) that 𝑑𝑌=𝑤𝑑𝑡𝑌+_𝑖Ω_𝑖𝑌_𝑖+𝑉𝜉,𝑑𝑁=−𝑤𝑑𝑡𝑁+_𝑖Ψ_𝑖𝑌_𝑖+Φ𝜉,𝑑𝑌_𝑖=−Ψ_𝑖𝑌−Ω_𝑖𝑁+_𝑗Ω_𝑖𝑗𝑌_𝑗+𝐹_𝑖𝜉,𝑑𝜉=−Φ𝑌−𝑉𝑁−_𝑖𝐹_𝑖𝑌_𝑖, (39) where Ω_𝑖𝑗=−Ω_𝑗𝑖, Ω_𝑖=𝜔_𝑖+𝑣_𝑖𝑑𝑡, 𝑉=𝑣𝑑𝑡. By exterior differentiation of (39), we get 𝑑Ω_𝑖=_𝑗Ω_𝑖𝑗∧Ω_𝑗+𝑤𝑑𝑡∧Ω_𝑖+𝑣𝑑𝑡∧𝐹_𝑖,𝑑𝑣∧𝑑𝑡=_𝑖Ω_𝑖∧𝐹_𝑖,𝑑𝐹_𝑖=_𝑗𝐹_𝑗∧Ω_𝑗𝑖−Ω_𝑖∧Φ−Ψ_𝑖∧𝑉,𝑑Ω_𝑖𝑗=_𝑘Ω_𝑖𝑘∧Ω_𝑘𝑗−Ω_𝑖∧Ψ_𝑗−Ψ_𝑖∧Ω_𝑗+𝐹_𝑖∧𝐹_j. (40) Since 𝑇^∗(𝑀^𝑛×𝑅)=𝑇^∗𝑀^𝑛⊕𝑇^∗𝑅, we have the following decomposition: Ω_𝑖𝑗=𝜔_𝑖𝑗+𝐿_𝑖𝑗𝑑𝑡,Ψ_𝑖=𝐴_𝑖+𝑢_𝑖𝑑𝑡,Φ=𝐶+𝑢𝑑𝑡,𝐹_𝑖=𝐵_𝑖+𝑏_𝑖𝑑𝑡, (41) where {𝑢_𝑖,𝑢,𝐿_𝑖𝑗,𝑏_𝑖} are local functions on 𝑀×ℝ. Using (40) and comparing the terms in 𝑇^∗𝑀∧𝑑𝑡, we get 𝜕𝜔_𝑖𝜕𝑡=_𝑗𝑣_𝑖,𝑗+𝐿_𝑖𝑗+𝐵_𝑖𝑗𝑣𝜔_𝑗+𝑤𝜔_𝑖, (42) where {𝑣_𝑖,𝑗} is the covariant derivative of ∑𝑣_𝑖𝐸_𝑖 with respect to 𝑔_𝑡. Here, we have used the notations of conformal invariants {𝐴_𝑖𝑗,𝐵_𝑖𝑗,𝐶_𝑖} for 𝑥_𝑡. By the same way, we get from (40) that 𝑏_𝑖=𝑒_𝑖(𝑣)+_𝑗𝐵_𝑖𝑗𝑣_𝑗,𝜕𝐹_𝑖=𝜕𝑡_𝑗𝑏_𝑖,𝑗+_𝑘𝐿_𝑖𝑘𝐵_𝑘𝑗+𝐴_𝑖𝑗𝑣−𝑣_𝑖𝐶_𝑗𝜔_𝑗+𝑢𝜔_𝑖, (43) where {𝑏_𝑖,𝑗} are covariant derivatives of ∑_𝑖𝑏_𝑖𝜔_𝑖. Using (42) and (43), we get 𝜕𝐵_𝑖𝑗𝜕𝑡+𝑤𝐵_𝑖𝑗=𝑏_𝑖,𝑗+_𝑘𝐿_𝑖𝑘𝐵_𝑖𝑘−𝐵_𝑖𝑘𝐿_𝑘𝑗−𝑣_𝑖𝐶_𝑗+𝐴_𝑖𝑗𝑣+𝑢𝛿_𝑖𝑗−_𝑘𝐵_𝑖𝑘𝑣_𝑘,𝑗+𝐵_𝑘𝑗𝑣,𝑏_𝑖,𝑗=𝑣_,𝑖𝑗+_𝑘𝐵_{𝑖𝑘,𝑗}𝑣_𝑘+𝐵_𝑖𝑘𝑣_𝑘,𝑗,𝑣_,𝑖𝑗=𝑑_𝑀𝑒_𝑖(𝑣)+_𝑗𝑒_𝑗(𝑣)𝜔_𝑗𝑖. (44) Therefore, we have 𝜕𝐵_𝑖𝑗𝜕𝑡+𝑤𝐵_𝑖𝑗=𝑣_,𝑖𝑗+_𝑘𝐿_𝑖𝑘𝐵_𝑖𝑘−𝐵_𝑖𝑘𝐿_𝑘𝑗+𝐵_{𝑖𝑗,𝑘}𝑣_𝑘+𝐴_𝑖𝑗−_𝑘𝐵_𝑖𝑘𝐵_𝑘𝑗+𝑢−_𝑘𝑣_𝑘𝐶_𝑘𝛿_𝑖𝑗,𝑛−1𝑛𝑤=_𝑖𝑗𝐵_𝑖𝑗𝑣_,𝑖𝑗+𝐴_𝑖𝑗−_𝑘𝐵_𝑖𝑘𝐵_𝑘𝑗𝑣. (45)(46) Now, we calculate the first variation of the following conformal volume functional: 𝑊(𝑡)=vol𝑔_𝑡=_𝑀𝑛𝜔₁∧⋯∧𝜔_𝑛=_𝑀𝑛𝑑𝑀_𝑡, (47) where 𝑑𝑀 is the volume for 𝑔_𝑡. From (42) and (46) we get 𝑊^′(𝑡)=_𝑖𝑀𝑛𝜔₁∧⋯∧𝜕𝜔_𝑖𝜕𝑡∧⋯∧𝜔_𝑛=𝑛_𝑀𝑛𝑤𝑑𝑀_𝑡=𝑛²𝑛−1_𝑀^𝑛_𝑖𝑗𝐵_𝑖𝑗𝑣_,𝑖𝑗+𝐴_𝑖𝑗−_𝑘𝐵_𝑖𝑘𝐵_𝑘𝑗𝑣𝑑𝑀_𝑡. (48) From the fact that the variation is admissible, we know that 𝑣_𝑖=0, 𝑣=0 and 𝑒_𝑖(𝑣)=0 on 𝜕𝑀^𝑛. It follows from (48) and Green’s formula that 𝑊^′(𝑡)=𝑛²×𝑛−1_{𝑀𝑛𝑖𝑗}𝐵_{𝑖𝑗,𝑖𝑗}+_𝑖𝑗𝐴_𝑖𝑗𝐵_𝑖𝑗−_𝑖𝑗𝑘𝐵_𝑖𝑘𝐵_𝑘𝑗𝐵_𝑖𝑗×𝑣𝑑𝑀_𝑡. (49) Thus, we have the following theorem.

Theorem 6. A space-like hypersurface 𝑥∶𝑀^𝑛→𝑀₁^𝑛+1(𝑐) is a Willmore hypersurface (i.e., a critical hypersurface to the conformal volume functional) if and only if _𝑖𝑗𝐵_{𝑖𝑗,𝑖𝑗}+_𝑖𝑗𝐴_𝑖𝑗𝐵_𝑖𝑗−_𝑖𝑗𝑘𝐵_𝑖𝑘𝐵_𝑘j𝐵_𝑖𝑗=0. (50)

Using (24), we can write the Euler-Lagrange equations as _𝑖𝐶_𝑖,𝑖+_𝑖𝑗1𝑅𝑛−1^𝑖𝑗−𝐴_𝑖𝑗𝐵_𝑖𝑗=0. (51)

Theorem 7. Any maximal (or zero mean curvature) space-like surface in Lorentz space forms is a Willmore surface.

Proof. Let 𝑥∶𝑀²→𝑅³₁ 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 _𝑖𝑗𝐴_𝑖𝑗𝐵_𝑖𝑗=_𝑖𝑗𝑒^−2𝜏𝐵_𝑖𝑗𝜏_𝑖𝜏_𝑗−𝜏_𝑖,𝑗+_𝑖𝑗𝑒^−𝜏𝐵_𝑖𝑗𝐵_𝑖𝑗𝐻. (52) Since we have the following relations of connections: 𝜔_𝑖𝑗=𝜃_𝑖𝑗+𝜏_𝑖𝜔_𝑗−𝜏_𝑗𝜔_𝑖, (53) a direct calculation implies that _𝑖𝐶_𝑖,𝑖=_𝑖𝑗𝑒^−2𝜏𝐵_𝑖𝑗𝜏_𝑖𝜏_𝑗−𝜏_𝑖,𝑗−𝑒^−3𝜏Δ_𝑀𝐻. (54) From (51), we have _𝑖𝐶_𝑖,𝑖−_𝑖𝑗𝐴_𝑖𝑗𝐵_𝑖𝑗=−𝑒^−3𝜏Δ_𝑀𝐻+𝑒^2𝜏_𝑖𝑗𝐵_𝑖𝑗𝐵_𝑖𝑗𝐻. (55)
If 𝑥 is a maximal space-like surface, that is, 𝐻=0 and 𝑅_𝑖𝑗=𝜅𝛿_𝑖𝑗, thus 𝑥 is Willmore.
Similarly, we can verify that maximal space-like surfaces in 𝑆³₁(1) and 𝐻³₁(−1) are also Willmore. Thus, we complete the proof of Theorem 7.

4. A Characteristic of CMC Hypersurfaces and 𝜅_𝑀=Constant

In this section, we consider space-like hypersurfaces with constant mean curvature and constant scalar curvature.

Proposition 8. Let 𝑥∶𝑀^𝑛→𝑀₁^𝑛+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 (1),𝐶=0,(2),𝐴=𝜇𝐵+𝜆𝑔, (56) where 𝜇, 𝜆 are constant.

Proof. First, we consider the space-like hypersurface 𝑥∶𝑀^𝑛→𝑅₁^𝑛+1. Since 𝐻 and 𝜅 are constant, then by the Gauss equation we have that 𝑒^2𝜏=𝑛||𝑛−1𝐼𝐼²−𝑛𝐻²=constant. (57) From (32), we get that 𝐶_𝑖=0,𝐵_𝑖𝑗=𝑒^−𝜏ℎ_𝑖𝑗−𝐻𝛿_𝑖𝑗,𝐴_𝑖𝑗=𝑒^−2𝜏12𝐻²−𝐻ℎ_𝑖𝑗. (58)
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 𝑀₁^𝑛+1(𝑐)=𝑆₁^𝑛+1(1) and 𝑀₁^𝑛+1(𝑐)=𝐻₁^𝑛+1(−1). Thus, we complete the proof of Proposition 8.

Theorem 9. Let 𝑥∶𝑀^𝑛→𝑀₁^𝑛+1(𝑐) be a space-like hypersurface without umbilical points. If conformal invariants of 𝑥 satisfy (1),𝐶=0,(2),𝐴=𝜇𝐵+𝜆𝑔. (59) Then 𝑥 is conformal equivalent to a space-like hypersurface with constant mean curvature and constant scalar curvature.

Proof. Since 𝐶=0, from (20), we can take the local orthonormal basis {𝐸₁,…,𝐸_𝑛} such that 𝐵_𝑖𝑗=diag𝑏₁,…,𝑏_𝑛,𝐴_𝑖𝑗=diag𝑎₁,…,𝑎_𝑛. (60) Since 𝐴=𝜇𝐵+𝜆𝑔, from structure equations (14) we get that 𝑑𝑁−𝜆𝑑𝑌−𝜇𝑑𝜉=0. (61) Taking exterior differentiation of (61), we get that 𝑑𝜆∧𝑑𝑌+𝑑𝜇∧𝑑𝜉=0. (62) Writing 𝑑𝜆=∑_𝑖𝜆_𝑖𝜔_𝑖, 𝑑𝜇=∑_𝑖𝜇_𝑖𝜔_𝑖, from (62) we get that _𝑖𝑗𝑘𝜆_𝑘𝛿_𝑖𝑗+𝜇_𝑘𝐵_𝑖𝑗𝜔_𝑘∧𝜔_𝑗𝑌_𝑖=0, (63) which implies that 𝜆_𝑘𝛿_𝑖𝑗+𝜇_𝑘𝐵_𝑖𝑗=𝜆_𝑗𝛿_𝑖𝑘+𝜇_𝑗𝐵_𝑖𝑘. (64) Set 𝑖=𝑗 in (64) and taking summation over 𝑖, we get that 𝜆_𝑘−1𝜇𝑛−1^𝑘𝑏_𝑘=0. (65) Set 𝑖=𝑗≠𝑘 in (65), from (60), we get that 𝜆_𝑘+𝜇_𝑘𝑏_𝑖=0,𝑖≠𝑘. (66) Combining (65) and (66), we get that 𝜇_𝑘𝑏_𝑖+1𝑏𝑛−1^𝑘=0,𝑖≠𝑘. (67)
If 𝜇_𝑖=0, 1≤𝑖≤𝑛, then 𝜆_𝑖=0, 1≤𝑖≤𝑛.
If 𝜇_𝑖≠0, for some 𝑖; we can assume that 𝜇₁≠0. Then, from (67), we have 𝑏_𝑖1=−𝑏𝑛−1¹,𝑖≠1. (68) Since tr(𝐵)=0 and |𝐵|²=(𝑛−1)/𝑛, thus we have 𝑏₁=𝑛−1𝑛,𝑏₂=⋯=𝑏_𝑛=−1𝑛. (69) Using 𝑑𝐵_𝑖𝑗+∑_𝑘𝐵_𝑘𝑗𝜔_𝑘𝑖+∑_𝑘𝐵_𝑖𝑘𝜔_𝑘𝑗=∑_𝑘𝐵_{𝑖𝑗,𝑘}𝜔_𝑘 and (69), we get that 𝜔_1𝑖=0,1<𝑖≤𝑛. (70) Thus, we have 𝑅_1𝑖1𝑖=0,1<𝑖≤𝑛,𝑎₂=⋯=𝑎_𝑛. (71) Since 𝑎_𝑖=𝜇𝑏_𝑖+𝜆, then from (71), we have 2𝜆+𝑛−2𝑛𝜇=𝑛−1𝑛²,2𝜆₁+𝑛−2𝑛𝜇₁=0. (72)(73) From (66), we have 𝜆₁+1𝑛𝜇₁=0. (74) From (73) and (74), we get that 𝜇₁=0. (75)
This is a contradiction; so 𝜇_𝑖=0, 1≤𝑖≤𝑛. Since 𝜇_𝑖=0, 𝜆_𝑖=0, 1≤𝑖≤𝑛, so 𝜆 and 𝜇 are constant.
From (61), we have 𝑑𝑁−𝜆𝑌−𝜇𝜉=0. (76) Therefore, we can find a constant vector 𝑒∈𝑅₂^𝑛+3 such that 𝑁−𝜆𝑌−𝜇𝜉=𝑒. (77) From (11) and (77), we get that ⟨𝑒,𝑒⟩=𝜇²−2𝜆,⟨𝑌,𝑒⟩=1. (78)
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 𝑒=−1,0,1=𝑒𝑇=𝑁−𝜆𝑌−𝜇𝜉𝑇. (79) Let 𝑥∶𝑀^𝑛→𝑅₁^𝑛+1 be a hypersurface that its conformal position vector is 𝑌=𝑌𝑇, then 𝑁=𝑁𝑇, 𝜉=𝜉𝑇, and 𝑒=𝑁−𝜆𝑌−𝜇𝜉. (80) Writing 𝑌=𝑒^𝜏⟨𝑥,𝑥⟩+12,⟨𝑥,𝑥,𝑥⟩−12,𝜉=−𝐻⟨𝑥,𝑥⟩+12,⟨𝑥,𝑥,𝑥⟩−12+𝑦_𝑛+1, (81) then from 𝑌,𝑒=1,𝜉,𝑒=−𝜇, (82) we obtain that 𝑒^𝜏=1,𝐻=𝜇. (83) Since 𝑌=((⟨𝑥,𝑥⟩+1)/2,𝑥,(⟨𝑥,𝑥⟩−1)/2), then 𝑔=⟨𝑑𝑥,𝑑𝑥⟩=𝐼. From (24) we have (𝐴)1tr=𝑛𝜆=𝑛2𝑛²𝜅−1. (84)
Since 𝜅_𝑀=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 𝑒=0,𝜇²−2𝜆=𝑒𝑇=𝑁−𝜆𝑌−𝜇𝜉𝑇. (85) Let 𝑥∶𝑀^𝑛→𝐻₁^𝑛+1(−1) be a hypersurface which its conformal position vector is 𝑌=𝑌𝑇; then 𝑁=𝑁𝑇, 𝜉=𝜉𝑇, and 𝑒=𝑁−𝜆𝑌−𝜇𝜉. (86) Writing 𝑌=𝑒^𝜏(𝑥,1), 𝜉=−𝐻(𝑥,1)+𝑦_𝑛+1, then from 𝑌,𝑒=1,𝜉,𝑒=−𝜇, (87) we obtain that 𝑒^𝜏=1𝜇²,−2𝜆𝐻=𝜇. (88) Since ⟨𝑑𝑥,𝑑𝑥⟩=(𝜇²−2𝜆)𝑔, 𝜅_𝑀=1𝜇²−2𝜆𝜅. (89)
Therefore, the mean curvature and scalar curvature of hypersurface 𝑥 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 𝑒=−2𝜆−𝜇²,0=𝑒𝑇=𝑁−𝜆𝑌−𝜇𝜉𝑇. (90) Let 𝑥∶𝑀^𝑛→𝑆₁^𝑛+1(1) be a hypersurface that its conformal position vector is 𝑌=𝑌𝑇; then 𝑁=𝑁𝑇, 𝜉=𝜉𝑇, and 𝑒=𝑁−𝜆𝑌−𝜇𝜉. (91) Writing 𝑌=𝑒^𝜏(1,𝑥), 𝜉=−𝐻(1,𝑥)+𝑦_𝑛+1, then from 𝑌,𝑒=1,𝜉,𝑒=−𝜇, (92) we obtain that 𝑒^𝜏=12𝜆−𝜇²,𝐻=𝜇. (93) Since ⟨𝑑𝑥,𝑑𝑥⟩=(2𝜆−𝜇²)𝑔, 𝜅_𝑀=12𝜆−𝜇²𝜅. (94)
Therefore, the mean curvature and scalar curvature of hypersurface 𝑥 are constant. Thus, we complete the proof the Theorem 9.

Corollary 10. Let 𝑥∶𝑀^𝑛→𝑀₁^𝑛+1(𝑐) be a space-like hypersurface without umbilical points. If conformal invariants of 𝑥 satisfy (1),𝐶=0,(2),𝐴=𝜆𝑔, (95) then 𝑥 is conformal equivalent to a maximal space-like hypersurface with constant scalar curvature in 𝐻₁^𝑛+1(−1).

Proof. From the process of proof of Theorem 9, 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 𝐻₁^𝑛+1(−1). In fact, using covariant derivative of ∇𝐵 and 𝐶=0, we have 12Δ|𝐵|²=_𝑖𝑗𝑘𝐵_{𝑖𝑗,𝑘}²+_𝑖𝑗Δ𝐵_𝑖𝑗=_𝑖𝑗𝑘𝐵_{𝑖𝑗,𝑘}²+𝑛−1𝑛²+(𝑛−1)𝑛tr(𝐴)+𝑛_𝑖𝑗𝑘𝐵_𝑖𝑘𝐵_𝑘𝑗𝐴_𝑗𝑖. (96) Since |𝐵|²=(𝑛−1)/𝑛 and 𝐴=𝜆𝑔, we get that _𝑖𝑗𝑘𝐵_{𝑖𝑗,𝑘}²+𝑛−1𝑛²+2(𝑛−1)𝜆=0. (97) 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 𝐻₁^𝑛+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 in 𝐻₁^𝑛+1(−1).

Example 11. Let 𝐻^𝑘𝑟₁=𝑢=𝑢₀,𝑢₁∈𝑅₁^𝑘+1∣⟨𝑢,𝑢⟩=−𝑟²₁,𝐻^𝑛−𝑘𝑟₂=𝑣=𝑣₀,𝑣₁∈𝑅₁^{𝑛−𝑘+1}∣⟨𝑣,𝑣⟩=−𝑟²₂, (98) where 𝑟²₁+𝑟²₂=1. Define space-like hypersurface 𝑥∶𝐻^𝑘𝑟₁×𝐻^𝑛−𝑘𝑟₂⟶𝐻₁^𝑛+1(),𝑥()=𝑢−1𝑢,𝑣₀,𝑣₀,𝑢₁,𝑣₁∈𝐻₁^𝑛+1(−1)⊂𝑅₂^𝑛+2. (99) Then, unit normal vector of 𝑥 is 𝑒_𝑛+1=((𝑟₂/𝑟₁)𝑢₀,−(𝑟₁/𝑟₂)𝑣₀,(𝑟₂/𝑟₁)𝑢₁,−(𝑟₁/𝑟₂)𝑣₁); thus, the induced metric and the second fundamental form of 𝑥 are, respectively, 𝐼=𝐼_𝑢+𝐼_𝑣𝑟,𝐼𝐼=²𝑟₁𝐼_𝑢−𝑟₁𝑟₂𝐼_𝑣,𝑟𝑛𝐻=𝑘²𝑟₁−()𝑟𝑛−𝑘¹𝑟₂,|𝐼𝐼|²𝑟=𝑘²𝑟₁²+()𝑟𝑛−𝑘¹𝑟₂². (100) 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 𝑘𝑟₂𝑟₁⁶+()𝑟3𝑘−2𝑛²𝑟₁⁴+()𝑟3𝑘−𝑛²𝑟₁²+𝑘=0. (101) Since 𝑟²₁+𝑟²₂=1, we get that 𝑟₁=−𝑛−𝑘𝑛,𝑟₂=−𝑘𝑛. (102)

Corollary 12. Space-like hypersurface 𝑥∶𝐻^𝑘(𝑟₁)×𝐻^𝑛−𝑘(𝑟₂)→𝐻₁^𝑛+1(−1) is Willmore if and only if 𝑟₁=−𝑛−𝑘𝑛,𝑟₂=−𝑘𝑛. (103)

Remark 13. Space-like hypersurface 𝑥∶𝐻^𝑘(𝑟₁)×𝐻^𝑛−𝑘(𝑟₂)→𝐻₁^𝑛+1(−1) is maximal if and only if 𝑟₁=−𝑘𝑛,𝑟₂=−𝑛−𝑘𝑛. (104) Thus, maximal hypersurfaces are not Willmore in general.

Theorem 14. Let 𝑥∶𝑀^𝑛→𝑀₁^𝑛+1(𝑐) be a Willmore space-like hypersurface without umbilical points. If conformal invariants of 𝑥 satisfy (1),𝐶=0,(2),𝐴=𝜇𝐵+𝜆𝑔, (105) then 𝑥 is conformal equivalent to the space-like hypersurface in 𝐻₁^𝑛+1(−1) with constant mean curvature and constant scalar curvature, and 2|ℎ|²𝐻=trℎ³+𝑛𝐻³,|ℎ|²−𝑛𝐻²≤𝑛, (106) where ℎ=(ℎ_𝑖𝑗). If |ℎ|²−𝑛𝐻²=𝑛, then 𝑥 is 𝐻^𝑘−𝑛−𝑘𝑛×𝐻^𝑛−𝑘−𝑘𝑛. (107)

Proof. Let 𝑥∶𝑀^𝑛→𝑀₁^𝑛+1(𝑐) be a Willmore space-like hypersurface without umbilical points. Let {𝐸₁,…,𝐸_𝑛} be the local orthonormal basis for 𝑔 such that 𝐵_𝑖𝑗=diag𝑏₁,…,𝑏_𝑛, (108) and corresponding principal curvatures are (𝑘₁,…,𝑘₂). Since (1),𝐶=0,(2),𝐴=𝜇𝐵+𝜆𝑔, (109) and since 𝑥 is Willmore, from (51), we get that 𝑛−1𝑛𝜇=_𝑖𝑏³_𝑖. (110) From (33), we have 2|ℎ|²𝐻=trℎ³+𝑛𝐻³. (111) By calculation of (1/2)Δ|ℎ|², we have 12Δ|ℎ|²=_𝑖𝑗𝑘ℎ_{𝑖𝑗,𝑘}²−𝑛²|∇𝐻|²+12_𝑖𝑗𝑅_{𝑖𝑗𝑖𝑗}𝑘_𝑖−𝑘_𝑗²=_𝑖𝑗𝑘ℎ_{𝑖𝑗,𝑘}²−𝑛²|∇𝐻|²+𝑛𝑐|ℎ|²−𝑛²𝑐𝐻²+|ℎ|⁴−𝑛𝐻_𝑖𝑘³_𝑖. (112) 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 |ℎ|²=constant,𝐻=constant. (113) From (112), we have 0=_𝑖𝑗𝑘ℎ_{𝑖𝑗,𝑘}²+𝑛𝑐|ℎ|²−𝑛²𝑐𝐻²+|ℎ|⁴−𝑛𝐻_𝑖𝑘³_𝑖. (114) Combining (111) and (114), we get that _𝑖𝑗𝑘ℎ_{𝑖𝑗,𝑘}²+𝑛𝑐|ℎ|²−𝑛𝐻²+|ℎ|²−𝑛𝐻²²=0. (115) Since |ℎ|²−𝑛𝐻²≥0 and from (115), we get that 𝑐=−1. (116) Thus, 𝑥 is a space-like hypersurface in 𝐻₁^𝑛+1(−1). From (115), we have _𝑖𝑗𝑘ℎ_{𝑖𝑗,𝑘}²+|ℎ|²−𝑛𝐻²|ℎ|²−𝑛𝐻²−𝑛=0. (117) Since |ℎ|²−𝑛𝐻²≥0, from (117), we have |ℎ|²−𝑛𝐻²≤𝑛. (118)
If |ℎ|²−𝑛𝐻²=𝑛, then ℎ_{𝑖𝑗,𝑘}=0; thus, the principal curvatures of 𝑥 are constant. It is well known that space-like isoparametric hypersurfaces in 𝐻₁^𝑛+1(−1) are either totally umbilical hypersurfaces or 𝐻^𝑘(𝑟₁)×𝐻^𝑛−𝑘(𝑟₂) (see [11]).
Thus, we complete the proof of Theorem 14.

Acknowledgments

Tongzhu Li is partially supported by Grant no. 10801006 of NSFC; Changxiong Nie is supported by Grant no. 1097055 of NSFC.