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Geometry
Volume 2013 (2013), Article ID 549602, 9 pages
http://dx.doi.org/10.1155/2013/549602
Research Article

Conformal Geometry of Hypersurfaces in Lorentz Space Forms

1Department of Mathematics, Beijing Institute of Technology, Beijing 100081, China
2Faculty of Mathematics and Computer Sciences, Hubei University, Wuhan 430062, China

Received 22 June 2013; Accepted 8 August 2013

Academic Editor: Anna Fino

Copyright © 2013 Tongzhu Li and Changxiong Nie. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let be a space-like hypersurface without umbilical points in the Lorentz space form . We define the conformal metric and the conformal second fundamental form on the hypersurface, which determines the hypersurface up to conformal transformation of . 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 with constant mean curvature and constant scalar curvature is Willmore, then is a hypersurface in .

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 (also see [2]). After that, the study of the Möbius geometry has been a topic of increasing interest (see [36]).

In this paper we study space-like hypersurfaces in the Lorentz space form 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 , which is the projectivized light cone in induced from (see [7, 8]). Using conformal compactification , 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 . 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 . Thus, isoparametric hypersurfaces in and 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 .

Let be the real vector space with the Lorenz inner product given by Let be a Lorentz space form. When , ; when , , and when , , where We denote by the cone in and by the conformal compactification space in : Let be the Lorentz group of keeping the Lorentz inner product invariant. Then, is a transformation group on defined by Topologically, is identified with the compact space , which is endowed by a standard Lorentz metric . has conformal metric:

and is the conformal transformation group of .

We define the following mappings (without ambiguity all denote by ): Using , we can regard hypersurfaces in as submanifolds in . A classical theorem states the following.

Theorem 1. Two hypersurfaces are conformal equivalent if and only if there exists such that .

Let be a space-like hypersurface without umbilical points; then, is a positive definite subbundle of . For any local lift of the standard projection , we get a local lift of , which is defined in an open subset of . Thus, is a local metric, which is conformal to the induced metric . We denote by and the Laplacian operator and the normalized scalar curvature with respect to the local positive definite metric , respectively. Then, we have the following.

Theorem 2. Let be a space-like hypersurface; then the 2-form is a globally defined conformal invariant. Moreover, is positive definite at any nonumbilical point of .

Proof. First, we prove that is well defined. Suppose that , are different lifts defined in open subsets and of . For the local positive definite metrics , we denote by the Laplace operator, by the gradient of a function and by the normalized scalar curvatures with respect to , respectively. Analogously for , we denote by the Laplace operator and by the normalized scalar curvatures. On , we have , where . Therefore, . By some computations, we have
Using these formula, it follows that
Next, we prove that is invariant under conformal transformations of . Let be a conformal transformation of , and we denote ; then, there is a acting on , and is a lift of 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 , therefore the 2-form is invariant under conformal transformations.
Now, let and take local lift ; 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 and . Thus; we complete the proof of Theorem 2.

Now, we assume that space-like hypersurface 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.

Theorem 3. Two space-like hypersurfaces are conformal equivalent if and only if there exists such that , where are the conformal position vectors of , respectively.

Let be a local orthonormal basis of with respect to with dual basis . Denote . We define

where is the Laplace operator of ; then, we have

We may decompose such that

where . We call the conformal normal bundle of , which is linear bundle. Let be a local section of and ; then, forms a moving frame in 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 -tensor, and conformal -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 , 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.

Theorem 4. Two space-like hypersurfaces and are conformal equivalent if and only if there exists a diffeomorphism 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 . But for time-like submanifolds, the situation is very different. In fact, for space-like submanifolds, the globally defined conformal invariant is positive definite or semipositive definite, and for time-like submanifolds, the globally defined conformal invariant is non-definite.

Next, we give the relations between the conformal invariants and isometric invariants of .

First, we consider space-like hypersurface in . Let be a space-like hypersurface without umbilical points. Let be an orthonormal local basis for the induced metric with dual basis . Let be a normal vector field of , and . Then, we have the first and second fundamental forms and the mean curvature . We may write ; ; . Denote by the Laplacian and the normalized scalar curvature for . By structure equation and Gauss equation of we get that For , there is a lift: So, we get It follows from (25) that Therefore, the conformal metric and conformal position vector of are as follows: Let ; then 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 .

Using the same methods, we can obtain relations between the conformal invariants and isometric invariants of and . We have the following unitied expression of the conformal invariants , , and : where for and for .

3. The First Variation of the Conformal Volume Functional

Let 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 be an admissible variation of such that for each small . For each , has the conformal metric . As in Section 2, we have a moving frame in and the Willmore functional . Let be a local basis for the conformal normal bundle of . Denote by and the differential operators on and , respectively. Then, we have By (31), we can find functions such that Since is a moving frame along , it follows from (37) and (38) that where , , . By exterior differentiation of (39), we get Since , we have the following decomposition: where are local functions on . Using (40) and comparing the terms in , we get where is the covariant derivative of 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 , and on . It follows from (48) and Green’s formula that Thus, we have the following theorem.

Theorem 6. A space-like hypersurface is a Willmore hypersurface (i.e., a critical hypersurface to the conformal volume functional) if and only if

Using (24), we can write the Euler-Lagrange equations as

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

Proof. Let be a space-like surface. Let be a local orthonormal basis of and a local normal vector field. From (32), we get that Since we have the following relations of connections: a direct calculation implies that From (51), we have
If is a maximal space-like surface, that is, and , thus is Willmore.
Similarly, we can verify that maximal space-like surfaces in and are also Willmore. Thus, we complete the proof of Theorem 7.

4. A Characteristic of CMC Hypersurfaces and

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

Proposition 8. Let be a space-like hypersurface without umbilical points. If the mean curvature and scalar curvature of are constant, then conformal invariants of satisfy where , are constant.

Proof. First, we consider the space-like hypersurface . Since and are constant, then by the Gauss equation we have that From (32), we get that
From (58), Let and ; then, we prove the formula (56).
Similarly, we can prove the formula (56) for and . Thus, we complete the proof of Proposition 8.

Theorem 9. Let be a space-like hypersurface without umbilical points. If conformal invariants of satisfy Then is conformal equivalent to a space-like hypersurface with constant mean curvature and constant scalar curvature.

Proof. Since , from (20), we can take the local orthonormal basis such that Since , from structure equations (14) we get that Taking exterior differentiation of (61), we get that Writing , , from (62) we get that which implies that Set in (64) and taking summation over , we get that Set in (65), from (60), we get that Combining (65) and (66), we get that
If , , then , .
If , for some ; we can assume that . Then, from (67), we have Since and , thus we have Using and (69), we get that Thus, we have Since , then from (71), we have From (66), we have From (73) and (74), we get that
This is a contradiction; so , . Since , , , so and are constant.
From (61), we have Therefore, we can find a constant vector such that From (11) and (77), we get that
To prove the theorem, we consider the following three cases.
Case  1. is light-like, that is, .
Case  2. is space-like, that is, .
Case  3. is time-like, that is, .
First, we consider Case  1; is ligh-tlike, that is, . Then, there exists a such that Let be a hypersurface that its conformal position vector is , then , , and Writing then from we obtain that Since , then . From (24) we have
Since , so the mean curvature and scalar curvature of hypersurface are constant.
Next, we consider Case  2; is space-like, that is, . Then, there exists a such that Let be a hypersurface which its conformal position vector is ; then , , and Writing , , then from we obtain that Since ,
Therefore, the mean curvature and scalar curvature of hypersurface are constant.
Finally, we consider Case  3; is time-like, that is, . Then, there exists a such that Let be a hypersurface that its conformal position vector is ; then , , and Writing , , then from we obtain that Since ,
Therefore, the mean curvature and scalar curvature of hypersurface are constant. Thus, we complete the proof the Theorem 9.

Corollary 10. Let be a space-like hypersurface without umbilical points. If conformal invariants of satisfy then is conformal equivalent to a maximal space-like hypersurface with constant scalar curvature in .

Proof. From the process of proof of Theorem 9, we have that is conformal equivalent to a maximal space-like hypersurface with constant scalar curvature. Next, we prove that is a maximal space-like hypersurface with constant scalar curvature in . In fact, using covariant derivative of and , we have Since and , we get that Thus, . From the process of proof of the Theorem 9, we have that is conformal equivalent to a maximal space-like hypersurface with constant scalar curvature in .

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 .

Example 11. Let where . Define space-like hypersurface Then, unit normal vector of is ; thus, the induced metric and the second fundamental form of are, respectively, Now, we assume that is Willmore. Since and are constant, from (33), we get that the Euler-Lagrange equation (51) is Since , we get that

Corollary 12. Space-like hypersurface is Willmore if and only if

Remark 13. Space-like hypersurface is maximal if and only if Thus, maximal hypersurfaces are not Willmore in general.

Theorem 14. Let be a Willmore space-like hypersurface without umbilical points. If conformal invariants of satisfy then is conformal equivalent to the space-like hypersurface in with constant mean curvature and constant scalar curvature, and where . If , then is

Proof. Let be a Willmore space-like hypersurface without umbilical points. Let be the local orthonormal basis for such that and corresponding principal curvatures are . Since and since is Willmore, from (51), we get that From (33), we have By calculation of , we have Since , , 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 From (112), we have Combining (111) and (114), we get that Since and from (115), we get that Thus, is a space-like hypersurface in . From (115), we have Since , from (117), we have
If , then ; thus, the principal curvatures of are constant. It is well known that space-like isoparametric hypersurfaces in are either totally umbilical hypersurfaces or (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.

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