ABSTRACT

The periodic s¹-equivariant hypersurfaces of constant mean curvature can be obtained by using the Lagrangians with suitable potential functions in the Berger spheres. In the corresponding Hamiltonian system, the conservation law is effectively applied to the construction of periodic s¹-equivariant surfaces of arbitrary positive constant mean curvature.

1. Introduction

W.-Y. Hsiang [1] investigated the rotation hypersurfaces of constant mean curvature in the hyperbolic or spherical -space. In [2], Eells and Ratto have constructed the rotation (-equivariant) minimal hypersurfaces in the unit 3-sphere with standard metric by using a certain first integral, which is invariant with respect to the rotation angle of generating curves on the orbit space. In [3], a family of -equivariant periodic CMC surfaces was constructed in the Berger spheres when the constant mean curvature (CMC) is a sufficiently small positive number, and it was cleared that the conserved quantity can be obtained by using the Lagrangian equipped with suitable potential function of the corresponding dynamical system with respect to the Hsiang-Lawson metric [1,4] on the orbit space via the Hamilton equation, where the rotation angle of generating curves can be regarded as “time”. We should remark that the corresponding Lagrangian has the vanishing potential when we construct the -equivariant minimal hypersurfaces. However, in case that we construct the -equivariant non-minimal CMC-hypersurface in the Berger sphere, the potential of the Lagrangian is a nonvanishing function. In Theorem 4.3, we determine the potential function of the Lagrangian which corresponds to the -equivariant CMCsurfaces immersed in the Berger sphere. As a result we can obtain a family of periodic -equivariant CMC surfaces in the Berger spheres when the constant mean curvature is an arbitrary positive number (Theorem 5.2).

2. Preliminaries

In [3], a generalized inner product on the unit 3- sphere was defined by

where, and , and are positive and nonnegative parameters, respectively. The Cartan hypersurface in the unit 4-sphere is covered by (via an 8-fold covering), whose metric is rescaled along the Hopf fibres and its metric on coincides with [5,6]. The family of metrics defined on contains this one as a special case. In particular is a left-invariant metric on and is called the Berger sphere with metric in case that. The Berger metrics are obtained from the canonical metric by multiplying the metric along the Hopf fiber by [7].

Throughout the paper we consider the Berger spheres. Here we summarize the notations which are used in the paper.

denotes the orbit space by -isometric - ction as follows.

As the parametrization of we use the following map:

stands for the orbital metric on :

is the volume function of orbits and is the Hsiang-Lawson metric on:

where

denotes a curve parametrized by arclength. And also and stand for the tension fields of with respect to the metrics and, respectively. The geodesic curvature at is defined by where denotes the unit normal vector field to.

3. S¹-Equivariant CMC-Immersion

For a curve, we consider an -equivariant map such that , where and are Riemannian submersions. Throughout the paper, we assume that is an -equivariant constant mean curvature immersion. Then we have

(1)

since

On the orbit space, the velocity vector field of a curve is given by the following component functions.

Lemma 3.1. The following formulas hold on .

(2)

(3)

where

and

Then using the formula (1) we have the following differential Equation (4) of generating curves which corresponds to the CMC-rotation hypersurfaces immersed in, since using Lemma 3.1 the geodesic curvature is given by

(4)

4. Conservation Laws

We consider a generating curve on such that and. Then we can consider the space of motion with

and time. Let be a Lagrangian on. Via the Legendre transformation we have the Hamiltonian on the phase space:

The conservation laws of our system imply the following Proposition 4.1. Let the Lagrangian on be the following form:

where is the Hsiang-Lawson metric on and is a potential function on the configuration space.

Then we have

(5)

where the conserved quantity in the formula represents the Hamiltonian of our system.

By means of the Hamilton equation (5), we shall determine the potential which corresponds to the -equivariant CMC surfaces immersed in via the differential Equation (4) of generating curves on the orbit space.

The direct computation yields the following Lemma 4.2. Assume that and are functions of

and. Then we have

(6)

where

As a consequence, we have the following Theorem 4.3. On our system, the Lagrangian and the Hamiltonian which correspond to the -equivariant CMC-H hypersurface immersed in can be determined as follows:

Proof. Using Lemma 4.2 and the differential equation of generating curves (4) we have

from which we obtain

Since is a constant mean curvature and

we can choose such as. Q.E.D.

5. Generating Curves for S¹-Equivariant CMC Surfaces

Let be a generating curve on such that and with the arc length. Then we set the following initial conditions:

The Hamilton equation (Theorem 4.3) implies that

from which we have

where

On the other hand, using the formulas

and

we have

Consequently we have the following Lemma 5.1. Under the initial conditions for generating curves which correspond to the CMC-H rotation hypersurfaces, we have

and

(resp.,) if and only if,

where

Assume that is an arbitrary positive number. In Lemma 5.1 we now choose such that

.

From Lemma 5.1, and there exists the value of such that decreases strictly until, where the value of equals to zero at, and takes a local minimum at. In fact, if does not take a local minimum, then we may assume that there exists

such that and

.

Then from the differential equation (4) of generating curves it follows that. On the other hand we obtain the following formula:

(7)

where

The formula (7) implies that

(8)

where

The formula implies that

from which we have

since.

Hence we see that is a positive number. Now if, then, which implies that and

, hence, which is a contradiction. Therefore, the value is not zero.

Consequently, since, from the formula (8)

we see that is not zero, which contradicts the assumption. Hence

takes a local minimum.

Thus we can continue as the curve satisfying the differential Equation (4) by the reflection. Let be the right hand side of (7). We can define by as follows:

Consequently we have the following Theorem 5.2. Let be an arbitrary positive number and choose such that. If is a rational number, then the corresponding -equivariant hypersurface is an immersed CMC-H torus in the Berger sphere. In particular, if is an integer, then this CMC-H torus is embedded.

Theorem 5.3. In the case, Then the corresponding -equivariant CMC-H hypersurface in the Berger sphere is an extended Clifford torus

where

Corollary 5.4. There exists an embedded minimal torus in the Berger sphere

6. Acknowledgements

I am grateful to Yoshihiro Ohnita and Junichi Inoguchi for their encouragement.

REFERENCES