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
![](https://www.scirp.org/html/5-5300333\ee0451d3-830e-49dd-b179-c54ea49d2bb4.jpg)
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.
![](https://www.scirp.org/html/5-5300333\3112d705-12c5-43d6-9d8d-d5c4eaddf2fd.jpg)
As the parametrization of
we use the following map:
![](https://www.scirp.org/html/5-5300333\531de402-d988-4f29-9fba-bd0af558525f.jpg)
stands for the orbital metric on
:
![](https://www.scirp.org/html/5-5300333\01c2f974-6cb6-4138-a574-a12fc55dff2e.jpg)
is the volume function of orbits and
is the Hsiang-Lawson metric on
:
![](https://www.scirp.org/html/5-5300333\4828c0ef-05de-4e62-ad03-ea7b31422e3e.jpg)
where
![](https://www.scirp.org/html/5-5300333\3d80e917-4317-4b33-bcf3-d8a626406af0.jpg)
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
![](https://www.scirp.org/html/5-5300333\881190a7-0bb5-4db2-a9b3-2542d636b835.jpg)
On the orbit space
, the velocity vector field of a curve
is given by the following component functions.
![](https://www.scirp.org/html/5-5300333\765c46f0-bc5d-440b-a0dc-21a003b25b37.jpg)
Lemma 3.1. The following formulas hold on
.
(2)
(3)
where
![](https://www.scirp.org/html/5-5300333\da2b4d91-1327-49f2-861b-31c137147a41.jpg)
and
![](https://www.scirp.org/html/5-5300333\eae569d4-9fbd-4b2d-b207-6c468c381c1c.jpg)
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 ![](https://www.scirp.org/html/5-5300333\ddbcafda-75d2-4bd9-bff4-8967c1f7d8c1.jpg)
and time
. Let
be a Lagrangian on
. Via the Legendre transformation we have the Hamiltonian
on the phase space
:
![](https://www.scirp.org/html/5-5300333\0ddc73b8-ead4-4ccf-adec-6d58ac410746.jpg)
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
![](https://www.scirp.org/html/5-5300333\6f9aa79f-f6bd-4e6e-959a-4cba18f69e98.jpg)
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:
![](https://www.scirp.org/html/5-5300333\72a42d77-7caa-4a5f-8f46-48ddb9e540c2.jpg)
Proof. Using Lemma 4.2 and the differential equation of generating curves (4) we have
![](https://www.scirp.org/html/5-5300333\7ef2c320-f46e-4b70-9239-918cf6271c04.jpg)
from which we obtain
![](https://www.scirp.org/html/5-5300333\ad6a2d08-5a55-404b-8f7b-a7e46065986e.jpg)
Since
is a constant mean curvature and
![](https://www.scirp.org/html/5-5300333\0787f2f8-d36f-44a0-ba43-17a81001e402.jpg)
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:
![](https://www.scirp.org/html/5-5300333\c74c7dd5-64c5-46a0-adeb-c5aee2fb68b4.jpg)
The Hamilton equation
(Theorem 4.3) implies that
![](https://www.scirp.org/html/5-5300333\729d6a86-f3c0-4e8b-8de8-31f0afc1a4eb.jpg)
from which we have
![](https://www.scirp.org/html/5-5300333\22cb76a9-4c93-4bbb-b5b7-6cfac5269713.jpg)
where
![](https://www.scirp.org/html/5-5300333\0035866d-9a25-4186-9548-30d63578afee.jpg)
On the other hand, using the formulas
![](https://www.scirp.org/html/5-5300333\bf665b4f-dba3-4c4d-95a7-b0a26a4ed8ab.jpg)
and
![](https://www.scirp.org/html/5-5300333\10ee167e-141e-49b5-961c-dabdb221b338.jpg)
we have
![](https://www.scirp.org/html/5-5300333\bc8a682e-f963-4b1c-8a06-b2b8563c9a5e.jpg)
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
![](https://www.scirp.org/html/5-5300333\162ad41e-aadf-4c9f-a1bd-f41d1c424441.jpg)
and
(resp.,
) if and only if,
![](https://www.scirp.org/html/5-5300333\5c1b4826-cc15-479c-abd9-9380a05aea31.jpg)
where
![](https://www.scirp.org/html/5-5300333\bc64a802-cbff-4ff0-8c5b-c758b90325bd.jpg)
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 ![](https://www.scirp.org/html/5-5300333\24413520-89e1-415a-9bc5-bd843a499e1e.jpg)
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
![](https://www.scirp.org/html/5-5300333\a0e3813e-0c50-40f4-bda8-27331bce7a13.jpg)
The formula (7) implies that
(8)
where
![](https://www.scirp.org/html/5-5300333\a3becfe0-b457-4377-8e13-3687f9c5958f.jpg)
The formula
implies that
![](https://www.scirp.org/html/5-5300333\8eefeedd-d44f-4c2f-a3a4-8ae6273fdf0d.jpg)
from which we have
![](https://www.scirp.org/html/5-5300333\c6bcf2c8-0718-45ce-bf46-b7cc477f1621.jpg)
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 ![](https://www.scirp.org/html/5-5300333\2a5989e6-e82b-40be-84ab-c02abcf453bf.jpg)
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:
![](https://www.scirp.org/html/5-5300333\b4d6c3c2-96e5-4689-bb5d-70a794e70fdf.jpg)
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
![](https://www.scirp.org/html/5-5300333\67e309fb-254a-496e-a1b0-a1458ae557c7.jpg)
where
![](https://www.scirp.org/html/5-5300333\eb79e114-34d3-4c33-aa72-f0ff632f188b.jpg)
Corollary 5.4. There exists an embedded minimal torus in the Berger sphere ![](https://www.scirp.org/html/5-5300333\1baa8478-c820-4d24-a8fd-d26d4815ff61.jpg)
![](https://www.scirp.org/html/5-5300333\639875f2-56d5-4c10-888f-f2bc3ef10fc4.jpg)
6. Acknowledgements
I am grateful to Yoshihiro Ohnita and Junichi Inoguchi for their encouragement.