Surfaces with Parallel Mean Curvature Vector in S^2xS^2 and H^2xH^2

Abstract

Two holomorphic Hopf differentials for surfaces of non-null parallel mean curvature vector in S^2xS^2 and H^2xH^2 are constructed. Using that, we classify those surfaces with vanishing Hopf differentials, and in particular spheres with parallel mean curvature vector.

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arXiv:0807.1808v2 [math.DG] 16 Oct 2008
SURFACES WITH PARALLEL MEAN CURVATURE
VECTOR IN S2× S2AND H2× H2
FRANCISCO TORRALBO AND FRANCISCO URBANO
Abstract. Two holomorphic Hopf differentials for surfaces of non-null
parallel mean curvature vector in S2×S2and H2×H2are constructed. A
1:1 correspondence between these surfaces and pairs of constant mean
curvature surfaces of S2× R and H2× R is established. Using that,
surfaces with vanishing Hopf differentials (in particular spheres with
parallel mean curvature vector) are classified and a rigidity result for
constant mean curvature surfaces of S2× R and H2× R is proved.
1. Introduction
Surfaces with constant mean curvature (CMC-surfaces) in three mani-
folds is a classic topic in differential geometry and it has been extensively
studied when the ambient manifold has constant curvature. In 2004, Abresh
and Rosenberg [1] studied CMC-surfaces in S2× R and H2× R where S2
(respectively H2) are the two-dimensional sphere (respectively the hyper-
bolic plane). They defined on such surfaces a holomorphic two-differential
which generalizes the classical Hopf differential defined for CMC-surfaces of
space forms. They also classified those CMC-surfaces with vanishing Hopf-
differential. In particular, they classified the orientable CMC-surfaces of
genus zero in S2× R and H2× R.
When the codimension of the surface is bigger than one, the natural gene-
ralization of these type of surfaces are the surfaces with parallel mean curva-
ture vector (in what follows PMC-surfaces). Although there are results for
codimension bigger than two, the most relevant ones are obtained when the
codimension is two. In 1971, Ferus [8] proved that a genus zero orientable
surface with (non-null) parallel mean curvature vector in a simply-connected
space form is a round sphere. In [5] and [13], Chen and Yau independently
classified all the surfaces with parallel mean curvature vector in space forms,
proving that they are CMC-surfaces of three dimensional umbilical hyper-
surfaces. Both results are based on the following fact: if H is the mean
curvature vector of the surface, as the dimension of the normal bundle is
two, it is possible to consider another parallel vector field in the normal bun-
dle˜H orthogonal to H with the same length and to define two holomorphic
Hopf differentials associated to H and˜ H.
Research partially supported by a MCyT-Feder research project MTM2007-61775 and
a Junta Andaluc´ ıa Grant P06-FQM-01642.
1

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2 FRANCISCO TORRALBO AND FRANCISCO URBANO
In 2000, Kenmotsu and Zhou [9] classified surfaces with parallel mean cur-
vature vector in the complex projective and the complex hyperbolic planes.
In this case, it is well known that there are not umbilical hypersurfaces of
these 4-manifolds, therefore there is not a method like in space forms to
construct surfaces of parallel mean curvature vector. The authors did not
use the existence of Hopf differentials. Instead, they reduced the classifica-
tion theorem, using a result by Ogata [12], to solve an O.D.E. system on the
surface. Using an analytic method, they classified these surfaces, proving
that there are few of them and they have a good behavior with respect to
the complex structure of the ambient space.
In this paper we study surfaces with parallel mean curvature vector in
S2×S2and H2×H2. In this case, although there are umbilical hypersurfaces
of the ambient space, only the totally geodesic ones (up to congruences S2×R
and H2×R) have constant mean curvature (see Proposition 1) and so CMC-
surfaces of S2×R and H2×R are surfaces with parallel mean curvature vector
in S2× S2and H2× H2respectively.
The most important idea in the paper is the construction of two holomor-
phic Hopf differentials on PMC-surfaces of S2×S2and H2×H2(see section
3) which generalize the Abresh-Rosenberg differential in the sense that if a
PMC-surface of S2× S2or H2× H2factorizes through a CMC-surface of
S2× R or H2× R, both Hopf differentials are equals and they coincide (up
to a constant) with the Abresh-Rosenberg differential (see Lemma 1). To
define these Hopf differentials we use the two K¨ ahler structures that these
4-manifolds have (see section 3).
In section 4 we prove the main results of the paper. Theorem 1 proves that
given a simply-connected Riemannian surface (Σ,g) there exists, up to con-
gruences, a 1:1 correspondence between PMC-isometric immersions of (Σ,g)
in S2×S2(respectively H2×H2) and pairs of CMC-isometric immersions of
(Σ,g) in S2× R (respectively H2× R). Moreover these two CMC-surfaces
are congruent if and only if the corresponding PMC-immersion factorizes
through a CMC-immersion of S2× R or H2× R. So the existence of full
PMC-immersions in S2×S2and H2×H2is deeply related to the rigidity of
CMC-immersions in S2× R and H2× R.
In Theorem 2 we classify an important family of surfaces of S2× S2and
H2×H2with parallel mean curvature vector: those that are Lagrangian sur-
faces with respect to some of the two K¨ ahler structures that these manifolds
have. Theorem 3 is the most important contribution of the paper, it classifies
the surfaces with parallel mean curvature vector with null extrinsic normal
curvature. In the classification it appears the CMC-surfaces of S2× R and
H2×R, the Lagrangian PMC-surfaces and a new family of PMC-surfaces in-
variant under 1-parameter groups of isometries of S2×S2and H2×H2which
are described in Proposition 4. This result allows us to classify the parallel
mean curvature surfaces with vanishing Hopf-differentials (Theorem 4) and
in particular the parallel mean curvature spheres of S2× S2and H2× H2
(Corollary 1).

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SURFACES WITH PARALLEL MEAN CURVATURE VECTOR3
In section 5, using Theorem 1 and the examples of Proposition 4, we
construct examples of CMC-surfaces in S2× R and H2× R. Among them
it is interesting to remark a two-parameter family of CMC-embedded tori
in S2× S1(Proposition 6). Moreover, Corollary 3 is a rigidity result for
CMC-surfaces of S2× R and H2× R. Finally, in section 6 we study general
properties of compact PMC-immersions in S2× S2and H2× H2.
The product of two Riemannian surfaces with different constant curva-
tures is not an Einstein manifold and this is a big problem in order to study
its PMC-surfaces. Following the ideas developed in this paper, on a PMC-
surface of the product of two Riemannian surfaces with constant curvatures
it is possible to define a holomorphic 2-differential, which coincides with the
sum of the two Hopf differentials when the constant curvatures are equals.
When these curvatures are opposite, this holomorphic differential was de-
fined in [11].
2. Preliminaries and examples
We denote by M2(ǫ), ǫ = 1,−1, the two-dimensional sphere S2= {x ∈
R3|x2
ture 1 when ǫ = 1 and the hyperbolic plane H2= {x ∈ R3|x2
−1, x3> 0} endowed with the canonical metric of constant curvature −1
when ǫ = −1. We denote by ω the K¨ ahler 2-form on M2(ǫ) and by J the
corresponding complex structure, i.e. ω(, ) = ?J ,? where ?, ? denotes the
metric of M2(ǫ).
If we consider M2(ǫ) × M2(ǫ) endowed with the product metric, which
will be also denoted by ?,?, then it is an orientable Einstein manifold with
scalar curvature 4ǫ. The orientation will be given by the 4-form π∗
where πj, j = 1,2 are the projections on the factors.
Along the paper we will consider M2(ǫ)× M2(ǫ) embedded isometrically
in R3× R3≡ R6when ǫ = 1 and in R3
the Lorentz-Minkowski 3-space.
Let Φ : Σ → M2(ǫ)×M2(ǫ) be an immersion of an oriented surface Σ. If
T⊥Σ is the normal bundle of Φ, then we have the orthogonal decomposition
Φ∗T(M2(ǫ) × M2(ǫ)) = TΣ ⊕ T⊥Σ.
Let¯∇ be the connection on Φ∗T(M2(ǫ)×M2(ǫ)) induced by the Levi–Civita
connection of M2(ǫ) × M2(ǫ) and let¯∇ = ∇ + ∇⊥be the corresponding
decomposition. If {e1,e2,e3,e4} is an oriented orthonormal local frame on
Φ∗T(M2(ǫ)×M2(ǫ)) such that {e1,e2} is an oriented frame on TΣ, then we
define the normal curvature K⊥of the immersion Φ by
K⊥= R⊥(e1,e2,e3,e4),
1+x2
2+x2
3= 1} endowed with the canonical metric of constant curva-
1+ x2
2− x2
3=
1ω ∧ π∗
2ω,
1× R3
1≡ R6
2when ǫ = −1, being R3
1
where R⊥is the curvature tensor of the normal connection ∇⊥. Also we
define the extrinsic normal curvature¯K⊥as the the function on Σ given by
¯K⊥=¯R(e1,e2,e3,e4),

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4 FRANCISCO TORRALBO AND FRANCISCO URBANO
where¯R is the curvature tensor of¯∇.
Definition 1. Let Φ : Σ → M2(ǫ)×M2(ǫ) be an immersion. We say that Φ
has non-null parallel mean curvature vector, from now on PMC-immersion,
if ∇⊥H = 0 and H is non-null. In such case, |H| is a positive constant.
Suppose that Φ : Σ → M2(ǫ)×M2(ǫ) is a PMC-immersion of an orientable
surface Σ. We can define another parallel normal vector field˜H as the only
one with |˜H| = |H| and {˜H,H} defining the same orientation on the normal
bundle as {e3,e4}. Because H is parallel, K⊥= 0 and hence the Ricci
equation of Φ is given by
|H|2¯K⊥= ?[AH,A˜ H]e1,e2?,
where Aξis the Weingarten endomorphism associated to a normal vector ξ.
In order to get examples of PMC-surfaces, we make use of the following
trivial fact: If Σ is a constant mean curvature surface of a totally umbilical
hypersurface with constant mean curvature of M2(ǫ) × M2(ǫ), then Σ has
parallel mean curvature vector as a surface of M2(ǫ) × M2(ǫ). Next propo-
sition describes the umbilical hypersurfaces with constant mean curvature
of M2(ǫ) × M2(ǫ).
Proposition 1. Let Ψ : N → M2(ǫ) × M2(ǫ) be a totally umbilical hyper-
surface with constant mean curvature. Then Ψ is totally geodesic and it is
locally congruent to the totally geodesic immersion:
ǫ = 1ǫ = −1
S2× R → S2× S2
(p,t) ?→ (p,(cost,sint,0))
Proof. Let η be a unit normal vector field of N in M2(ǫ) × M2(ǫ), ˆ σ the
second fundamental form of Ψ andˆH the mean curvature. As Ψ is totally
umbilical we have that
ˆ σ(v,w) =ˆH?v,w?η,
AsˆH is constant, we obtain that
H2× R → H2× H2
(p,t) ?→ (p,(0,sinht,cosht)).
∀v,w ∈ TN.
(∇ˆ σ)(x,v,w) = 0,
∀x,v,w ∈ TN.
Now the Codazzi equation of Ψ says that
¯R(x,v,w,η) = 0,
∀x,v,w ∈ TN.
Using this property for suitable vectors and that dimN = 3, it is easy to
get that one of the two components of η vanishes, and so, up to an isometry
of M2(ǫ) × M2(ǫ), we can take η = (0,η2).
If Ψ = (Ψ1,Ψ2), then Ψ andˆΨ = (Ψ1,−Ψ2) are an orthogonal reference
in the normal bundle of M2(ǫ) × M2(ǫ) in R6or R6
taking into account that ?ˆΨ∗(v),η? = −?v,η? = 0, we have that
v · η = −ˆAηv = −ˆHv.
2. So for any v ∈ TN,

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SURFACES WITH PARALLEL MEAN CURVATURE VECTOR5
So the map η +ˆHΨ : N → R6is a constant A = (A1,A2) ∈ R3× R3≡ R6,
and henceˆHΨ1= A1and η2+ˆHΨ2= A2. As N is a 3-manifold and Ψ
an immersion, Ψ1cannot be a constant and soˆH = 0 which implies that
Ψ is totally geodesic. Now the second equation says that η2= A2and so
?Ψ2,A2? = ?Ψ,η? = 0 with |A2| = |η2| = 1. This proves that Ψ2(N) is a
geodesic of S2or H2and the proof finishes.
?
As a consequence of this result we obtain that
CMC-surfaces of M2(ǫ) × R are surfaces of M2(ǫ) × M2(ǫ)
with parallel mean curvature vector.
Other examples of PMC-surfaces of M2(ǫ)×M2(ǫ) can be constructed in the
following way: given two regular curves α : I → M2(ǫ) and β : I′→ M2(ǫ)
then
Φ : I × I′→ M2(ǫ) × M2(ǫ)
Φ(t,s) = (α(t),β(s))
is an immersion of the surface I × I′whose mean curvature vector is given
by
H =kα
2(Jα′,0) +kβ
2(0,J˙β),
where′(respectively ˙) stands for the derivative with respect to t (respec-
tively s), kαand kβare respectively the curvatures of α and β and we have
assumed that |α′| = |˙β| = 1. So we obtain that Φ has parallel mean curva-
ture vector if and only if α and β are curves of constant curvature. In that
case, 4|H|2= k2
geodesics. It is interesting to remark that the induced metric on I × I′by
Φ is flat.
Taking into account the curves of constant curvature of S2and H2we
have that the above examples are, up to congruences, open subsets of the
following family of complete and embedded PMC-surfaces:
α+ k2
β, and hence Φ is minimal if and only if α and β are
Example 1. When ǫ = 1, the tori product of two geodesic circles
Ta,ˆ a= {(x,y) ∈ S2× S2/x3= a, y3= ˆ a},
whose mean curvature satisfy 4|H|2=
When ǫ = −1, we obtain three topological families of examples
1. the tori product of two geodesic circles
0 ≤ a ≤ ˆ a < 1, a2+ ˆ a2> 0,
ˆ a2
1−ˆ a2.
a2
1−a2+
ˆTa,ˆ a= {(x,y) ∈ H2× H2/x3= a, y3= ˆ a},
whose mean curvature satisfy 4|H|2=
2. the cylinders product of a geodesic circle and a hypercycle
1 < a ≤ ˆ a,
a2
a2−1+
ˆ a2
ˆ a2−1and |H|2> 1/2,
Ca,b= {(x,y) ∈ H2× H2/x3= a, y1= b},
whose mean curvature satisfy 4|H|2=
b ≥ 0, a > 1,
b2+1and |H|2> 1/4,
a2
a2−1+
b2