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.