Compact minimal surfaces in the Berger spheres

Abstract

We construct compact arbitrary Euler characteristic orientable and non-orientable minimal surfaces in the Berger spheres. Besides we show an interesting family of surfaces that are minimal in every Berger sphere, characterizing them by this property. Finally we construct, via the Daniel correspondence, new examples of constant mean curvature surfaces in the products S^2 x R, H^2 x R and in the Heisenberg group with many symmetries. Comment: 16 pages, 2 figures

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COMPACT MINIMAL SURFACES IN THE BERGER SPHERES
FRANCISCO TORRALBO
Abstract. We construct compact arbitrary Euler characteristic orientable
and non-orientable minimal surfaces in the Berger spheres. Besides we
show an interesting family of surfaces that are minimal in every Berger
sphere, characterizing them by this property. Finally we construct, via
the Daniel correspondence, new examples of constant mean curvature
surfaces in S2× R, H2× R and the Heisenberg group with many sym-
metries.
1. Introduction
The study of minimal surface in space forms is a classical topic in dif-
ferential geometry. It is well know that there is no compact minimal sur-
face in the Euclidean 3-space and in the hyperbolic space. On the con-
trary, Lawson [Law70] showed that every compact surface but the pro-
jective plane can be minimally immersed in the 3-sphere. His construc-
tion is based on the following fact: if a minimal surface contains a geo-
desic arc it can be continued as an analytic minimal surface by geodesic
reflection (see Definition 1). Using this property he constructed compact
minimal surfaces by reflecting the Plateau solution over a nice geodesic
polygon. This method has been successfully applied to other spaces and
other kinds of surface, like constant mean curvature ones (see, for exam-
ple [KPS88, K89, GB93]).
Besides the spaces form, the most regular 3-manifolds are the homo-
geneous Riemannian ones. Between them, the compact type examples,
which are expected to have compact minimal surfaces as in the space form
case, are the Berger spheres. The author has been study the simplest con-
stant mean curvature surfaces, in particular the minimal ones, in these
family of homogeneous riemannian manifolds (cf. [Tor10] and section 3).
Besides some studies about the stability of the constant mean curvature
surfaces has been done (cf. [TU10]).
In this paper we deal with the following problem: the construction of
compact minimal surfaces in the Berger spheres. These 3-manifolds are
homogeneous Riemannian with isometry group of dimension 4 which are,
roughly speaking, 3-spheres endowed with a family of deformation met-
rics of the round one (see section 2 for details). The main trouble we are
2000 Mathematics Subject Classification. Primary 53C42; Secondary 53C30.
Key words and phrases. Surfaces, minimal, homogeneous 3-manifolds, Berger spheres.
Research partially supported by a MCyT-Feder research project MTM2007-61775 and a
Junta Andalucía Grant P06-FQM-01642.
1
arXiv:1007.1072v1 [math.DG] 7 Jul 2010

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2FRANCISCO TORRALBO
going to find is the following: the reflection over a geodesic of the Berger sphere
could not be an ambient isometry, only a few of them are.
First section is devoted to present in detail the family of Berger spheres
and its geodesics and it shows what kind of geodesics satisfy that the
reflection over them is an ambient isometry (cf. Lemma 1). The second
section characterizes a new family of minimal examples as the only ones
that are minimal with respect to every Berger metric (see Proposition 1).
Section 4 and 5 developed the main result (cf. Theorems 1, 2 and Corol-
lary 1) in the paper:
Every compact surface but the projective plane can be minimally
immersed in any Berger sphere. Furthermore, in the oriented case
the immersion could be chosen without self intersection.
Finally, in section 6 we construct, as an application of the previous re-
sults and taking into account the Daniel correspondence [D07], complete
constant mean curvature surfaces in the product spaces S2×R and H2×R
(where S2stands for the 2-sphere and H2for the hyperbolic plane) and in
the Heisenberg group with many symmetries (see Proposition 2). In the
product space case we are able to characterize which kind of symmetries
these surfaces have (see Lemma 3).
This paper was prepared during the author’s visit to IMPA (Brazil).
The author wish to thank Harold Rosenberg for its valuable comments
and suggestion and to Ildefonso Castro for the description of the minimal
surfaces in the 3-sphere invariant by a Killing field (see (3.4)).
2. Preliminaries
A Berger sphere is a usual 3-sphere S3= {(z,w) ∈ C2: |z|2+ |w|2= 1}
endowed with the metric
g(X,Y) =4
κ
?
?X,Y? +
?4τ2
κ
− 1
?
?X,V??Y,V?
?
where ?, ? stands for the usual metric on the sphere, V(z,w)= J(z,w) =
(iz,iw), for each (z,w) ∈ S3and κ, τ are real numbers with κ > 0 and
τ ?= 0. For now on we will denote the Berger sphere (S3, g) as S3
note that if κ = 4τ2then S3
κ ?= 4τ2then the group of isometries of S3
where J is the matrix J =
0 J1
different connected components U+(2) and U−(2) given by
U+(2) = {A ∈ O(4) : AJ = JA},
and each of one is homeomorphic to the unitary group U(2). U+(2),
after identifying it with the unitary group, acts in the usual way over
pair of complex numbers while U−(2) acts by conjugation, that is, given
A = (z1 z2
b(κ,τ). We
b(κ,τ) is, up to homotheties, the round sphere. If
b(κ,τ) is {A ∈ O(4) : AJ = ±JA}
?
J1 0
?
and J1=
?0 −1
1 0
?. Iso(S3
b(κ,τ)) has two
U−(2) = {A ∈ O(4) : AJ = −JA}
w1w2) ∈ U−(2) then
A(z,w) = (z1¯ z + z2¯ w,w1¯ z + w2¯ w)

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COMPACT MINIMAL SURFACES IN THE BERGER SPHERES3
Taking into account the above relation between the standard metric on
the sphere and the Berger metric, it is not difficult to get the next formula
that links the Levi–Civita connection ∇ of the round sphere to ∇b, the one
associated to the Berger metric:
?4τ2
where J is the complex structure of C2, that is J(z,w) = (iz,iw), and ()⊥
denotes the tangential component to the sphere.
The Hopf fibration Π : S3
2-sphere of radius 1/√κ,
?
is a Riemannian submersion whose fibers are geodesics. The vertical unit
Killing field is given by ξ =
As the Berger spheres are homogeneous Riemannian manifolds to know
all the geodesics it is enough to describe them at a fixed point, for example
(1,0) ∈ S3. So, let w =
unit vector, then the geodesic starting at (1,0) with speed w is given by
γθ,ϕ(s) = (z1(s),z2(s)) where
?κ − 4τ2
z2(s) = expi
4τ
(2.1)
∇b
XY = ∇XY +
κ
− 1
?
[?Y,V?(JX)⊥+ ?X,V?(JY)⊥]
b(κ,τ) → S2(κ), where S2(κ) stands for the
Π(z,w) =
2
√κ
z ¯ w,1
2(|z|2− |w|2)
?
,
κ
4τV.
?
iκ
4τcosθ,
√κ
2sinθeiϕ?
∈ T(1,0)S3
ban arbitrary
z1(s) = expi
4τ
cosθ s
??
cos
?sinθ
?λ√κ
2
s
?
?λ√κ
+ i2τ
√κ
cosθ
λ
?
sin
?λ√κ
2
s
??
?
ϕ +κ − 4τ2
cosθ s
λ
sin
2
s
and λ2= sin2θ +4τ2
horizontal if its tangent vector is orthogonal to V and vertical if its tan-
gent vector is co-linear with V. This terminology comes from the Hopf
fibration, i.e., the vertical geodesics are the fibers of this Riemannian sub-
mersion. Notice that the horizontal and vertical geodesics are great circles.
The ones starting at (1,0) are given by:
?
v(s) =
eiκ
,
v?(0) =
κcos2θ (cf. [Rak85]). We will say that a geodesic is
hϕ(s) =
cos
?√κ
4τs,0
2s
?
?
,sin
?√κ
2s
?
κ
4τ(i,0)
eiϕ
?
,
h?
ϕ(0) =
√κ
2(0,eiϕ),
?
(2.2)
Remark 1. It is important to make the following remarks:
(1) Given two horizontal geodesics starting at (1,0), hϕ1and hϕ2, the
angle between them is ϕ1− ϕ2.
(2) The length of every vertical geodesic is 8τπ/κ, while the length of
every horizontal geodesic is 4π/√κ.
Definition 1. A geodesic reflection across a geodesic γ of S3
map that sends a point p to its “opposite” point on a geodesic through p
which meets γ orthogonally. More precisely, if α is a geodesic that meet
b(κ,τ) is the

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4FRANCISCO TORRALBO
orthogonally γ at s = 0 and α(s0) = p then the geodesic reflection of p
with respect to γ is the point α(−s0).
Lemma 1. The geodesic reflection across an horizontal or a vertical geodesic is an
isometry of S3
b(κ,τ).
Proof. As the Berger spheres are homogeneous Riemannian manifolds it
is enough to check that the geodesic reflection across the vertical geodesic
through (1,0) and all the horizontal geodesic through (1,0) are isometries.
In the first case it is easy to see that such transformation is given by
rv(z,w) = (z,−w) which is clearly an isometry of the Berger sphere.
In the second case we fix first the horizontal geodesic given by h0(cf.
(2.2)). In this special case it is not difficult to see that the geodesic reflec-
tion across h0is given by R0(z,w) = (¯ z, ¯ w) which is an isometry. Now,
as the rotation of angle θ around the vertical geodesic v through (1,0) is
ρ(z,w) = (z,eiθw) we can recover the geodesic reflection across any hori-
zontal geodesic through (1,0) by conjugation with this rotation. Therefore
the geodesic reflection across the horizontal geodesic hϕ(cf. (2.2)) is given
by Rϕ(z,w) = (¯ z,e2iϕ¯ w) which is an isometry. This finishes the proof.
?
3. New minimal surfaces in the Berger spheres
In [Tor10] it is studied the simplest examples of constant mean curva-
ture surfaces in the Berger sphere, indeed the rotationally ones, that is, the
constant mean curvature surfaces invariant by the 1-parameter group of
isometries given by t →
minimal ones:
• The equator {(z,w) ∈ S3
minimal sphere, up to ambient isometries, in S3
• the Clifford torus {(z,w) ∈ S3
• a 1-parameter family of examples, call unduloids because they looks
like euclidean unduloids. Some of them produces tori and, even
more, for some Berger spheres with small bundle curvature τ (with
respect to a fixed κ) there exist embedded ones (cf. [Tor10, Remark
3.2]).
The first two examples are in fact minimal surfaces in every Berger
sphere, that is, they are minimal surfaces in S3
τ ?= 0. So a natural question arises: Does exist more surfaces that are simulta-
neously minimal in all the Berger spheres? This section is devoted to answer
positively this question providing a new 1-parameter family of minimal
surfaces in the Berger spheres (see Proposition 1).
Firstly we are going to related the mean curvature of an arbitrary sur-
face in a Berger sphere with its mean curvature as a surface in the usual
3-sphere.
?
1 0
0 eit
?
. Among them it was found the following
b(κ,τ) : Im(z) = 0}, which is the only
b(κ,τ),
b(κ,τ) : |z|2= |w|2=1
2},
b(κ,τ) for every κ > 0 and
Lemma 2. Let Φ : Σ → S3be an immersion of a surface Σ. Then the mean
curvatures H and Hbof Φ in the round sphere (S3,?, ?) and the Berger sphere

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COMPACT MINIMAL SURFACES IN THE BERGER SPHERES5
S3
b(κ,τ) respectively are related by
√κ
Hb=
2
1
?
1−?1−
κ
4τ2
?ν2
?
H +
?1−
κ
4τ2
??∇ν,V?
4τ2
2?1−?1−
κ
?ν2?
?
where ν = ?N,V?, N is a unit normal vector field of Σ in (S3,?, ?) and V(z,w)=
(iz,iw).
Proof. First if N and Nbare two unit normal vector field of Φ in S3and
S3
√κ
?
where ν = ?N,V?.
Now taking into account (2.1) and the above relation between the nor-
mal vector fields we get
?4τ2
?
?4τ2
where {e1,e2} is an orthonormal reference with respect to ?, ? that diago-
nalize the shape operator of Φ : Σ → (S3,?, ?), λiare the principal curva-
tures and f : Σ → R is the function given by f =√κ
Finally, as −2Hb= ∑2
yields the result.
b(κ,τ) respectively then they are related by the equality:
Nb=
21−?1−
κ
4τ2
?ν2
?N − ν
?
1−
κ
4τ2
?
V?,
κ
4fg(∇b
eiNb,ej) =
κ
− 1
?
?ei,V??ej, JN?+
?4τ2
− 1
− λi
δij+
κ
− 1
?
?ei,V??ej,V??
+
−
κ
??ej,V?ei(ν),
?
2
?
1−?1−
κ
4τ2
?ν2.
i,j=1gijg(∇b
eiNb,ej) a straightforward computation
?
Thanks to the above result if Σ is a minimal surface in the round sphere
then it is minimal in some Berger sphere S3
sphere) if and only if ?∇ν,V? = 0. The equator {(z,w) ∈ S3: Im(z) = 0}
and the Clifford torus {(z,w) ∈ S3: |z|2= |w|2=1
so they are, as we already knew, minimal surfaces in every Berger sphere.
Next proposition classifies all the surfaces that are simultaneously minimal
in every Berger sphere.
b(κ,τ) (in fact, in every Berger
2} satisfy this condition
Proposition 1. Let Φ : Σ → S3be an immersion of a surface Σ. Suppose that Φ
is a minimal immersion with respect to any Berger metric. Then Φ is congruent
to a piece of one of the immersion Φc: R2→ S3given by:
Φc(s,t) =
?
cos(s)eict, sin(s)eit?
,
where c ∈ R.
Remark 2.
(1) If c = 0 we get the equator {(z,w) ∈ C2: Im(z) = 0}.