O(p + 1) x O(p + 1)-Invariant hypersurfaces with zero scalar curvature in euclidean space

Abstract

We use equivariant geometry methods to study and classify zero scalar curvature O(p + 1) x O(p + 1)-invariant hypersurfaces in R2p+2 with p > 1.

equivariant geometry; scalar curvature

O(p+1)´O(p+1) - Invariant Hypersurfaces with Zero Scalar Curvature in Euclidean Space

JOCELINO SATO^*

Universidade Federal do Ceará, UFC, Departamento de Matemática,

Campus do Pici, 60455-760 Fortaleza, CE, Brazil

Manuscript received on November 20, 1999; accepted for publication on February 8, 2000;

presented by MANFREDO DO CARMO

ABSTRACT

We use equivariant geometry methods to study and classify zero scalar curvature O(p+1)´O(p+1)-invariant hypersurfaces in R^2p+2 with p>1.

Key words: equivariant geometry, scalar curvature.

1. INTRODUCTION

The methods of equivariant geometry have been applied successfully by many authors to obtain and classify explicit examples of hypersurfaces, with a given condition on the r-th mean curvature, that are invariant by the action of an isometry group (see, for instance, Hsiang et al. 1983, Hsiang 1982, do Carmo & Dajczer 1983, Bombieri et al. 1969, Alencar 1993).

O. Palmas (Palmas 1999), resuming a work started initially by T. Okayasu (Okayasu 1989) and using ideas contained in Alencar, 1993, published a work in which he approaches the hypersurfaces with zero scalar curvature in R^2p+2 , invariant by the action of the group O(p+1)´O(p+1). In his article, Palmas studied only the case p = 1.

The objective of this work is to announce and give an sketch of proof of a classification theorem for the case . The orbit space of the action is the set and the invariant hypersurfaces are generated by curves , the so called profile curves, that satisfy the following diferential equation

(1)

In order to study the profile curves of such hypersurfaces we proceeded as in Alencar 1993, analyzing the trajectories of an associated vector field . Each trajectory of is associated to a family of hypersurfaces generated by profile curves , determined by up to homothety. The profile curves in the orbit space of these hypersufaces are one of the following types:

A) is one of the following half-straight line

where t ³ and (see figure 1);

B) is regular, intersects orthogonally one of the half-axes or and asymptotizes one of the half-straight lines in case A), when or (see figure 1);

C) is the union of two curves and being a singularity. The curves do not intesect the boundary of the orbit space, and asymptotizes the half-straight lines of the case A, when (see figure 1 );

D) is regular and does not intersect the boundary of the orbit space and asymptotizes both half-straight lines of the case A, when (see figure 1)

Fig. 1
- Profile curves.

We will denote by C_a and the cones generated by the half-straight lines of type A.

The main result of this work is the theorem below classifing O(p+1)´O(p+1)-invariant hypersurfaces according to their profile curves.

CLASSIFICATION THEOREM. O(p+1)´O(p+1)-invariant hypersurfaces in with and zero scalar curvature belong to one of the following classes:

1. cones with a singularity in the origin of

   

   (type A).

2. hypersurfaces that have one orbit of singularities and that are asymptotic to both the cones C

   _a

   e

   

   (type C).

3. regular hypersurfaces that are asymptotic to the cone C

   _a

   (type B).

4. regular hypersurfaces that are asymptotic to the cone

   

   (type B).

5. regular hypersurfaces that are asymptotic to both cones C

   _a

   and

   

   (type D).

As a corollary we obtain the following result.

THEOREM A. Let be an O(p+1)´O(p+1)-invariant hypersurface in R^2p+2 complete and with zero scalar curvature. Then is generated by a curve of type B or D. Moreover

i) If

is generated by a curve of type B, then

is embedded and asymptotic to one of the cones C_a or ;

ii) If

is generated by a curve of type D, then

is embedded and asymptotic to both of the cones C_a and .

The cones C_a and , generated by the half-straight lines in case A are characterized in the following theorem:

THEOREM B. If is an O(p+1)´O(p+1)-invariant hypersurface in R^2p+2 with zero scalar curvature whose profile curve makes a constant angle with the -axes then is one of the cones C_aor .

This work is organized as follows. In section 2 we reduce the study of the profile curves of the invariant hypersurfaces in , with zero scalar curvature, to the study of the trajectory of a vector field . Then we use the qualitative theory of ordinary differential equations, together with a geometric analysis of the behavior of , to obtain a description of its trajectories.

In section 3, we present sketches of the proofs of the theorems announced above.

2. ANALYSIS OF THE VECTOR FIELD X

The regular curves satisfing the equation are invariant by homotheties and, therefore, for each solution of (1) we have a family of invariant hypersurfaces with zero scalar curvature, generated by the curves So we can apply the method developed in (Bombieri et al. 1969) to study the corresponding differential equation. Also note that, if a curve is a solution of equation (1), then is also a solution.

Without loss of generality, we may assume that the curves are parametrized by arc length. Therefore, when we obtain

(2)

Proceeding as in Bombieri et al. 1969 we introduce the parameters

(3)

which are invariant by the homothety Assuming , we rewrite equation (1) as the system

We associate to this system the vector field in the -plane.

Since our orbit space is the region , we need information just for , corresponding to the region in the -plane .We observe that is bounded, -periodic in both variables and invariant by a translation of . So, it is enough to analyse it in the interval .

In order to characterize the phase portrait of the field we make a geometric study of its behaviour. This study gives us information about the increasing and decreasing intervals of the coordinates and of an orbit , the types of singularities that presents and a transversality of on special curves. This tranversality supplies barriers for the possible behaviors of those orbits of .

These informations, together with the tubular flow theorem and Poincaré-Bendixson's theorem allow us to prove the following proposition, where we use the notation:

PROPOSITION 1. The trajectories of are defined for all values of . In the region their possible behaviors is one of the following:

1) is a vertical trajectory with -limit and -limit or a vertical trajectory with -limit and -limit or still a vertical trajectory with -limit and -limit

2) is a vertical half-trajectory with -limit or a vertical half-trajectory with -limit

3) is a trajectory in with -limit and -limit going through the points of where

4) is a connection of saddle points contained in the region with -limit and -limit

5) is a connection of saddle points contained in the region with -limit and -limit .

6) is a connection of saddle points contained in the region with -limit and -limit .

7) is a connection of saddle points contained in the region with -limit and -limit

8) is a trajectory contained in the region with -limit and -limit .

9) is an orbit, or part of one, obtained by a translation of of one of the orbts given in the itens 1-8 .

3.O(p+1)´O(p+1)-INVARIANT HYPERSURFACES IN R^2p+2

The hypersurfaces of type A (item 1 of the Classification theorem) are given by the cones e and characterized in Theorem B, whose proof consists in to use that, if is a solution with then it satisfies the equation This, together with the fact that is parametrized by arc length, give us the result.

Theorem A follows from the Classification theorem, Lemma 1 and Remark 1 below.

LEMMA 1. Let be a trajectory with -limit and -limit . Let be the associated profile curve. Then intersects the segment exactly once, so intersects the diagonal exactly once. Therefore, does not possess self-intersections and the hypersurface generated by is embedded and complete.

REMARK 1. If is a profile curve associated to a connection of saddle points, then is a graph over one of the axes x or y, and intersects it orthoganally. Therefore the hypersurface generated by is embedded and complete.

The proof of the Classification theorem is a consequence of the Proposition 1, together with the remark below:

REMARK 2. For we have and so we can see the profile curve as a graph (or union of graphs when ) has singularities) of a function or We will assume without loss of generality, that In this case, equation (2) tells us that there are singularities at the zeros of the equation

They correspond to the coordinates with

ACKNOWLEDGEMENT

This work is part of the author's doctoral thesis at the Department of Mathematics of the Federal University of Ceará (Fortaleza - CE). He would like to thank his adviser, Prof. Luquésio P. M. Jorge, for his guidance. A detailed account of the material presented in this announcement will appear elsewhere.

REFERENCES

ALENCAR H. 1993. Minimal Hypersyfaces in R^2m invariant by SO(m) ´ SO(m), Trans Amer Math Soc, 337(1): 129-141.

BOMBIERI E, DE GIORGI E & GIUSTI E. 1969. Minimal cones and the Berstein problem, Inventiones Math, 7: 243-269.

DO CARMO MP & DAJCZER M. 1983. Rotational hypersurfaces in spaces of constant curvature, Trans Amer Math Soc 277(2): 685-709.

HSIANG WY. 1982. Generalized rotational hypersurfaces of constant mean curvature in the Euclidean spaces I, Journal Diff Geom, 17: 337-356.

HSIANG WY, TENG ZH & YU WC. 1983. New examples of constant mean curvature immersions of (2k – 1)-spheres into Euclidean 2k-space, Annals of Math, 117: 609-625.

OKAYASU T. 1989. O(2) ´ O(2)-invariant hypersurfaces with constant negative scalar curvature in E⁴, Proc. of the AMS, 107: 1045-1050.

PALMAS O. 1999. O(2) ´ O(2)-invariant hypersurfaces with zero scalar curvature, to appear in Archiv der Mathematik.

• BOMBIERI E, DE GIORGI E & GIUSTI E. 1969. Minimal cones and the Berstein problem, Inventiones Math, 7: 243-269.
• DO CARMO MP & DAJCZER M. 1983. Rotational hypersurfaces in spaces of constant curvature, Trans Amer Math Soc 277(2): 685-709.
• HSIANG WY. 1982. Generalized rotational hypersurfaces of constant mean curvature in the Euclidean spaces I, Journal Diff Geom, 17: 337-356.
• HSIANG WY, TENG ZH & YU WC. 1983. New examples of constant mean curvature immersions of (2k 1)-spheres into Euclidean 2k-space, Annals of Math, 117: 609-625.

Publication Dates

• Publication in this collection
  07 Aug 2000
• Date of issue
  June 2000

History

• Accepted
  08 Feb 2000
• Received
  20 Nov 1999