Table Of ContentMINIMAL LAGRANGIAN SURFACES IN S2 S2
×
6
0
ILDEFONSOCASTROANDFRANCISCOURBANO
0
2
Abstract. We deal with the minimal Lagrangian surfaces of the Einstein-
n
Ka¨hlersurfaceS2×S2,studyinglocalgeometric propertiesandshowingthat
a
J they canbe locallydescribedas Gauss maps of minimalsurfaces inS3 ⊂R4.
We also discuss the second variation of the area and characterize the most
6
relevantexamplesbytheirstabilitybehaviour.
2
]
G
D 1. Introduction
.
h The theory of minimal surfaces in three-dimensional Riemannian manifolds of
at constant sectional curvature is a classical topic in differential geometry which has
m been developed in large. Recently, many geometers are interested in minimal sur-
faces of 3-dimensional manifolds of type Σ R, where Σ is a Riemann surface like
[
the 2-sphere S2 or the hyperbolic space H2×. As an illustrative example we refer to
1
[MR] and references there in.
v
When the ambient space M is a four-dimensional Riemannian manifold of con-
7
3 stant sectional curvature, there are different approaches to the theory of minimal
6 surfaces (see among others [B]) that is specially rich if M is an Einstein-K¨ahler
1 surface. In this case one can study their minimal surfaces according with their
0
behaviour with respect to the K¨ahler structure, considering in this way important
6
families of surfaces: complex surfaces (which are alwaysminimal), Lagrangiansur-
0
/ faces, totally real surfaces, etc. (see for instance [W]). The most relevant ambient
h
space in this setting is the complex projective plane, whose minimal surfaces have
t
a been extensively studied.
m
Besidesthecomplexprojectiveplane,thereisanotheronlyHermitiansymmetric
: space of compact type and complex dimension 2: the Einstein-K¨ahlersurface S2
v S2. In this article we start the study of their minimal Lagrangian surfaces. I×n
i
X section 2, we give a brief introduction to the geometry of S2 S2 as well as their
×
r Lagrangian surfaces, appearing a function C on the surface (called the associated
a
Jacobian)whichisstudiedindepthandwillplayanimportantrolealongthepaper.
In section 3 we show two recipes for constructing Lagrangian surfaces of S2 S2:
as graphs of area-preserving diffeomorphisms of the sphere S2 (see Example×1) or
as Gauss maps of certain surfaces of the Euclidean space R4 (see Example 2).
In section 4 we first classify the compact Lagrangian surfaces of S2 S2 with
×
non null parallelmean curvature vector (Theorem 1), by defining two holomorphic
differentialsonthe surfacethatallowtoprovethattheassociatedJacobianC isan
isoparametricfunction. WealsostudygeneralpropertiesoftheminimalLagrangian
1991 Mathematics Subject Classification. Primary53C42,53B25; Secondary53A05,53D12.
Keywordsandphrases. Lagrangiansurfaces,minimalsurfaces,HamiltonianstableLagrangian
surfaces,IndexofminimalLagrangiansurfaces,ExtremalmetricontheKleinbottle.
ResearchpartiallysupportedbyaMEC-FedergrantMTM2004-00109.
1
2 ILDEFONSOCASTROANDFRANCISCOURBANO
surfacesofS2 S2,showingthatthezeroesofthenonnegativefunction1 4C2are
× −
the zeroes of a holomorphic 2-differential defined on the surface (see Proposition 5
for some consequences of this fact). Secondly we classify the minimal Lagrangian
surfacesofS2 S2 withconstantGausscurvatureandproviderigidityresultsabout
×
the Gauss curvature of these surfaces (Theorem 2). The Gauss maps of orientable
minimal surfaces of S3 R4 are weighty examples of minimal Lagrangiansurfaces
ofS2 S2 (see section4⊂.3). InTheorem3 we provea localconverseof this fact, by
establ×ishingthatanysimply-connectedminimalLagrangiansurfaceofS2 S2 such
that 1 4C2 has not zeroes is the Gauss map of a minimal surface of S3× R4.
− ⊂
A significantfactthatalsomotivates the presentpaper is thatthe only example
of extremal metric on a Klein bottle, i.e. a critical metric for the functional first
eigenvalue of the Laplacian, recently discovered by D. Jakobson, N. Nadirashvili
and I. Polterovich [JNP], can be described as the induced metric on the minimal
LagrangianKlein bottle embedded in S2 S2 defined by
×
B= ((x,z),(y,w)) S2 S2/2x=y, (√zw)= (√zw) ,
{ ∈ × ℜ ℑ }
where we consider S2 = (x,z) R C/x2+ z 2 = 1 . As a consequence, B is a
{ ∈ × | | }
Hamiltonian stable minimal Lagrangiansurface, i.e. a minimal Lagrangiansurface
stable for the area under Hamiltonian deformations of S2 S2.
×
In section 5 we study the second variation of the area functional for compact
minimal Lagrangiansurfaces of S2 S2 proving the following uniqueness results:
×
The totally geodesic Lagrangian sphere
M = (x, x) S2 S2/x S2
0
{ − ∈ × ∈ }
is theuniquestableminimal Lagrangian compact surfaceof S2 S2.
×
The totally geodesic Lagrangian sphere M , the totally geodesic La-
0
grangian torus
T= (x,y) S2 S2/x =y =0
1 1
{ ∈ × }
and the minimal Lagrangian Klein bottle B are the unique Hamil-
tonian stable minimal Lagrangian compact surfaces of S2 S2 with
×
genus g 2, when the surface is orientable, and with Euler charac-
≤
teristic χ 0, when the surface is non orientable.
≥
The index of an unstable minimal Lagrangian compact surface of
S2 S2 is at least 2 and it is 2 only for the totally geodesic La-
×
grangian torus T.
2. Lagrangian surfaces in S2 S2
×
Let S2 be the unit sphere in the Euclidean space R3 endowed with its standard
Euclideanmetric , anditsstructureofRiemannsurfacegivenbyJ v =x v,for
x
any v T S2, x h Si2, where stands for the vectorial product in R3. Its×K¨ahler
x
∈ ∈ ×
2-form is the area 2-form ω defined by ω (v,w) = J v,w =det x,v,w for any
0 0 x
v,w T S2. h i { }
x
W∈eendowS2 S2 withtheproductmetric(alsodenotedby , )andtheproduct
× h i
complex structure J given by
J (v)=(J v ,J v )=(x v ,y v ),
(x,y) x 1 y 2 1 2
× ×
MINIMAL LAGRANGIAN SURFACES IN S2 S2 3
×
for any v = (v ,v ) T (S2 S2), (x,y) S2 S2, which becomes S2 S2 in
1 2 (x,y)
∈ × ∈ × ×
a K¨ahler surface. Its K¨ahler 2-form is ω =π ω +π ω where π , i=1,2, are the
1∗ 0 2∗ 0 i
projections of S2 S2 onto S2.
The groupofis×ometries ofS2 S2 is the subgroupofthe orthogonalgroupO(6)
×
given by
A 0 0 A
(2.1) , /A,B O(3) .
0 B B 0 ∈
(cid:26)(cid:18) (cid:19) (cid:18) (cid:19) (cid:27)
The subgroup of holomorphic (resp. subset of antiholomorphic) isometries of S2
S2 is defined by the additional conditions A,B SO(3) (resp. detA = detB ×=
1). We point out that there are isometries of ∈S2 S2 neither holomorphic nor
− ×
antiholomorphic.
Considering S2 S2 S5(√2) R6 and denoting by σ˜ the second fundamental
form of S2 S2 int×o R6⊂, we have t⊂hat
×
(2.2) σ˜(v,w)=( v ,w x, v ,w y),
1 1 2 2
−h i −h i
where v =(v ,v ),w =(w ,w ) T (S2 S2), (x,y) S2 S2 R3 R3. It is
1 2 1 2 (x,y)
∈ × ∈ × ⊂ ×
clear that σ˜ satisfies
(2.3) σ˜(Jv,Jw)=σ˜(v,w),
what implies that the mean curvature vector H˜ of S2 S2 in R6 is given by
×
(2.4) 2H˜ = (x,y), (x,y) S2 S2.
(x,y)
− ∀ ∈ ×
In particular, S2 S2 is a minimal hypersurface of S5(√2). Using now the Gauss
equation of S2 ×S2 into R6, we obtain that S2 S2 is an Einstein-K¨ahler surface
× ×
of constant scalar curvature 4.
As an Hermitian symmetric space, we can identify S2 S2 with the Grassmann
manifold G+(2,4) of oriented 2-planes in the Euclidean×space R4 in the following
way. Let Λ2R4 = v w/v,w R4 R6 be the space of 2-vectors in R4 endowed
{ ∧ ∈ }≡
with the Euclidean metric , given by
hh ii
v w,v w = v,v w,w v,w w,v ,
′ ′ ′ ′ ′ ′
hh ∧ ∧ ii h ih i−h ih i
for any v,w,v ,w R4. We define the subspaces Λ2R4 of Λ2R4 generated by the
′ ′
unit vectors ∈ ±
1
E1 = (e e e e ),
1 2 3 4
± √2 ∧ ± ∧
1
E2 = (e e e e ),
1 3 4 2
± √2 ∧ ± ∧
1
E3 = (e e e e ),
1 4 2 3
± √2 ∧ ± ∧
where e ,e ,e ,e is anorientedorthonormalframeofR4,anddenote byS2 the
1 2 3 4
unit sp{heres in the}3-spaces Λ2R4. ±
If v ,v isanorientedort±honormalframeofaplaneP G+(2,4)andtakev ,
1 2 3
v in{such a}way that v ,v ,v ,v is an oriented orthono∈rmal frame in R4, then
4 1 2 3 4
{ }
the map
G+(2,4) S2 S2
−→ +× −
1 1
P (v v +v v ), (v v v v ) ,
7→ √2 1∧ 2 3∧ 4 √2 1∧ 2− 3∧ 4
(cid:18) (cid:19)
defines a diffeomorphism.
4 ILDEFONSOCASTROANDFRANCISCOURBANO
Let Φ = (φ,ψ) : Σ S2 S2 be an immersion of a surface Σ and denote by
→ ×
g =φ , +ψ , the induced metric. The immersionΦ is saidto be Lagrangian if
∗ ∗
h i h i
Φ ω =0, i.e. φ ω +ψ ω =0. This means that
∗ ∗ 0 ∗ 0
0= JdΦ (v),dΦ (w) = Jdφ (v),dφ (w) + Jdψ (v),dψ (w) ,
p p p p p p
h i h i h i
for any p Σ and v,w T Σ.
p
If Φ =∈(φ,ψ) : Σ ∈S2 S2 is an immersion of an oriented surface with area
→ ×
2-form ω , we can define the Jacobians of φ and ψ by
Σ
φ ω =Jac(φ)ω , ψ ω =Jac(ψ)ω .
∗ 0 Σ ∗ 0 Σ
Hence, when Σ is oriented, Φ is Lagrangian if and only if Jac(φ)= Jac(ψ). We
−
will call the function
(2.5) C :=Jac(φ)= Jac(ψ)
−
the associated Jacobian of the oriented Lagrangian surface Σ. Moreover, if Σ is
compact, we have that
1
(2.6) deg(φ)= deg(ψ)= Cω :=d.
Σ
− 4π
ZΣ
We will call this number d the degree of the oriented compact Lagrangian surface
Σ.
Ingeneral(whenΣisnotnecessarilyorientable),thesquare(φ ω )2and(ψ ω )2
∗ 0 ∗ 0
are well-defined and so it is also well-defined the function
C2 = Jdφ (e ),dφ (e ) 2 = Jdψ (e ),dψ (e ) 2 =
p 1 p 2 p 1 p 2
h i h i
= dφ (e ) dφ (e )2 = dψ (e ) dψ (e )2,
p 1 p 2 p 1 p 2
| × | | × |
where e ,e is an orthonormal basis of (T Σ,g). If R¯ denote the curvature ope-
1 2 p
rator o{f S2 }S2, from (2.2) it is easy to prove that R¯(e ,e ,e ,e ) = 2C2. So the
1 2 2 1
×
Gauss equation of Φ can be written as
σ 2
(2.7) K =2C2+2H 2 | | ,
| | − 2
where K is the Gauss curvature of Σ, H the mean curvature of Φ and σ the
second fundamental form of Φ. Thanks to the Lagrangian character of Φ, the
complex structure J defines an isomorphism between the tangent bundle to Σ and
the normal bundle of Φ. Therefore, dΦ (e ),dΦ (e ),JdΦ (e ),JdΦ (e ) is an
p 1 p 2 p 1 p 2
orthonormalframe at T (S2 S2),{and using (2.3) and (2.4), we obtain t}hat
Φ(p)
×
2
(2.8) 2Φ(p)=4H˜ =2 σ˜(dΦ (e ),dΦ (e )).
Φ(p) p i p i
−
i=1
X
Since Φˆ = (φ, ψ) is normal to S2 S2 into S5(√2), we have that H˜,Φˆ = 0. If
− × h i
we use (2.2), (2.8) and the fact that e ,e is an orthonormal frame in the above
1 2
{ }
equality, we prove that
(2.9) dφ (e )2+ dφ (e )2 = dψ (e )2+ dψ (e )2 =1.
p 1 p 2 p 1 p 2
| | | | | | | |
In particular, the maps φ,ψ : Σ S2 satisfy that the ranks of dφ and dψ at any
→
pointofΣ arealwayspositive. Hence, if C 0 then the ranksofdφ anddψ atany
point must be necessarily 1 and so both fu≡nctions φ and ψ define curves in S2. It
is clear that the product of two spherical curves is a Lagrangiansurface in S2 S2
×
with null associated Jacobian. In conclusion, we get the following result.
MINIMAL LAGRANGIAN SURFACES IN S2 S2 5
×
Proposition 1. Let φ : Σ S2 S2 be a Lagrangian immersion. Then C 0 if
→ × ≡
and only if Φ is locally the product immersion of two spherical curves
I I S2 S2
′
× −→ ×
(t,s) (α(t),β(s)).
7→
If we consider the product of two great circles of S2, we obtain a Lagrangian
surface congruent to
(2.10) T= (x,y) S2 S2/x =y =0 S1 S1.
1 1
{ ∈ × }≡ ×
Of course, the associated Jacobian of T is null and its degree is zero. In addition,
T is totally geodesic and flat.
We also get from (2.5) and (2.9) that
(2.11) dφ (e )2 dφ (e )2 2+4 dφ (e ),dφ (e ) 2 =
p 1 p 2 p 1 p 2
| | −| | h i
= dψp(e1)2 (cid:0) dψp(e2)2 2+4 dψp(e(cid:1)1),dψp(e2) 2 =1 4C2(p),
| | −| | h i −
which im(cid:0)plies that C2(p) 1/4(cid:1)and the equality holds if and only if φ and ψ are
≤
conformal maps at p Σ. If C2 1/4, we have that φ and ψ are conformal maps
∈ ≡
and (2.9) implies that
(2.12) φ∗ , =ψ∗ , =g/2.
h i h i
In particular, φ and ψ are local diffeomorphisms. Thus, the immersion Φ can be
locally reparametrizedby
Φ=(i,F):U S2 S2 S2,
⊂ −→ ×
where i is the inclusion of U into S2 and F is a diffeomorphism from U onto
F(U). But (2.12) means that F is the restriction to U of an isometry A of S2.
Moreover,as A ω +ω =0 we have that detA= 1. So our immersion is locally
∗ 0 0
holomorphically congruent to (i, i):U S2 S2−by the holomorphic (see (2.1))
− → ×
I 0
isometry . We summarize this in the following result.
0 A
(cid:18) − (cid:19)
Proposition 2. If Φ : Σ S2 S2 is a Lagrangian immersion then C2 1/4
→ × ≤
and C2 1/4 if and only if Φ(Σ) is congruent to an open subset of the Lagrangian
surface M≡ of S2 S2 defined by
0
×
(2.13) M = (x, x) S2 S2/x S2 .
0
{ − ∈ × ∈ }
ItiseasytocheckthatM istotallygeodesicwithC 1/2,d=1anditsGauss
0
≡
curvature is constant K 1/2.
In [CN] it was proved≡that the totally geodesic Lagrangian surfaces of S2 S2
×
are congruent to open subsets of M or T.
0
We study nowtheorientablecompactLagrangiansurfacesembeddedinS2 S2.
×
Proposition 3. Let Φ:Σ S2 S2 be a Lagrangian immersion of an orientable
→ ×
compact surface Σ. If Φ is an embedding, then either the genus of Σ is zero and
the degree of Φ is 1 or the genus of Σ is one and the degree of Φ is zero.
±
Proof. In that follows, we use the notation of [W, Section 3]. We denote by [Σ]
H (Σ,Z) the fundamental homology class of Σ and by Φ [Σ]# H2(S2 S2,Z∈)
2
∗ ∈ ×
6 ILDEFONSOCASTROANDFRANCISCOURBANO
the Poincar´e dual of Φ [Σ] H (S2 S2,Z). The self-intersection number of Φ is
2
∗ ∈ ×
defined by
I =(Φ [Σ]# Φ [Σ]#)([S2 S2]).
Φ
∗ ∪ ∗ ×
Since Φ is an embedding, it is given by I = χ(T Σ), where χ(T Σ) is the Euler
Φ ⊥ ⊥
number of the normal bundle. But
(Φ [Σ]# Φ [Σ]#)([S2 S2])=(Φ [Σ]#)(Φ [Σ]# [S2 S2])
∗ ∪ ∗ =×(Φ [Σ]#)(Φ∗[Σ]), ∗ ∩ ×
∗ ∗
and, using that Φ is a Lagrangianimmersion, χ(T Σ)= χ(Σ)=2(g 1), where
⊥
− −
g is the genus of Σ. Hence, we obtain that
(2.14) (Φ [Σ]#)(Φ [Σ])=2(g 1).
∗ ∗ −
On the other hand, if x is a point of S2, we know that [S2 x ],[ x S2]
H (S2 S2,Z) are generators of H (S2 S2,Z) Z2. Then it×is{w}ell-{kn}ow×n tha∈t
2 2
× × ≡
their Poincar´e duals satisfy:
([S2 x ]#)([S2 x ])=0, ([S2 x ]#)([ x S2])=1,
([ x×{S}2]#)([S2×{x}])=1, ([ x×{S}2]#)([{x}×S2])=0.
{ }× ×{ } { }× { }×
As Φ [Σ]=deg(φ)[S2 x ]+deg(ψ)[ x S2], we obtain that
∗ ×{ } { }×
([Φ [Σ]#)([Φ [Σ])=2deg(φ)deg(ψ).
∗ ∗
Using that deg(ψ) = deg(φ), from (2.14) we finally get that g 1 = deg(φ)2,
what proves the Propo−sition. − − (cid:3)
Remark 1. The totally geodesic Lagrangiansurfaces M and T defined in (2.13)
0
and (2.10) show that the result proved in Proposition 3 is the best possible one.
Inthefollowingresult,weshowthatM andTaretheonlyexamplesofcompact
0
Lagrangiansurfaces in S2 S2 with constant associated Jacobian.
×
Proposition 4. There do not exist Lagrangian immersions of compact surfaces in
S2 S2 with constant C2 (0,1/4).
× ∈
Proof. LetΦ=(φ,ψ):Σ S2 S2beaLagrangianimmersionofacompactsurface
→ ×
such that C2 is constant, 0 < C2 < 1/4. First, since C2 does not have zeroes, the
rankofdφanddψ is2atanypointofΣ. Soφandψ arelocaldiffeomorphismsand
the compacity of Σ implies that they are in fact diffeomorphisms. Therefore Σ is a
sphere,ΦmustbeanembeddingandwecanconsiderthatC (whichiswell-defined)
is a constant C (0,1/2). Then Proposition 3 says that the degree d = 1 and so
∈
Area(Σ)C =4π from (2.6).
Next we prove that the Gauss curvature of Σ is also constant, concretely K =
2C2: Using the definition (2.5) of C, we have that
2C(p)= JdΦ (e ),dΦˆ (e ) , p Σ,
p 1 p 2
h i ∈
where e ,e is an oriented orthonormal frame in T Σ and Φˆ = (φ, ψ). If v
1 2 p
{ } − ∈
T Σ, then
p
(2.15) 2v(C)= σ(v,e ),JdΦˆ (e ) σ(v,e ),JdΦˆ (e ) =0.
2 p 1 1 p 2
h i−h i
Using that the trilinear form (u,v,w) σ(u,v),Jw is fully symmetric for a
7→ h i
Lagrangiansurface, we can write
σ(e ,e )=λJe +ηJe , σ(e ,e )=ηJe +µJe , σ(e ,e )=µJe +νJe ,
1 1 1 2 1 2 1 2 2 2 1 2
MINIMAL LAGRANGIAN SURFACES IN S2 S2 7
×
withλ,η,µ,ν R. Putting in (2.15)v =e andv =e respectivelyandusing that
1 2
∈
e ,e is an orthonormalframe, we get that
1 2
{ }
(µ λ) dφ (e ),dφ (e ) +η(dφ (e )2 dφ (e )2)=0,
p 1 p 2 p 1 p 2
− h i | | −| |
(ν η) dφ (e ),dφ (e ) +µ(dφ (e )2 dφ (e )2)=0.
p 1 p 2 p 1 p 2
− h i | | −| |
Takinginto accountthat 1 4C2(p)>0,from(2.11)the lastequationsimply that
−
µ(µ λ)= η(ν η). But this is equivalent to σ 2 = 4H 2 at any arbitrary point
− − | | | |
p. The Gauss equation (2.7) now gives K =2C2. The Gauss-Bonnettheorem says
that2Area(Σ)C2 =4π,butweprovedthatArea(Σ)C =4π. ThuswegetC =1/2,
which is a contradiction to the hypothesis. (cid:3)
3. Examples of Lagrangian surfaces.
Inthissection,weemphasizetwointerestingwaysofconstructionofLagrangian
surfaces in S2 S2.
×
3.1. Graphs of area-preserving diffeomorphisms. Let F :U S2 S2 be a
smooth map defined on an open set U of S2 and consider the graph⊆of F→,
Φ:U S2 S2
−→ ×
Φ(x)=(x,F(x)).
We have that Φ is Lagrangianif and only if ω +F ω =0. In particular, F must
0 ∗ 0
be a local diffeomorphism.
When U = S2, F must be a diffeomorphism and the last equation means that
F preserves the area of S2. As a summary:
−
Example 1. If F : S2 S2 is an area preserving diffeomorphism, the graph of
→
F:
−
M= (x, F(x)) S2 S2/x S2 ,
{ − ∈ × ∈ }
is a Lagrangian surface of S2 S2. In particular, the totally geodesic Lagrangian
surface M defined in (2.13) i×s the graph of the antipodal map I :S2 S2.
0
− →
On the other hand, if Φ = (φ,ψ) :Σ S2 S2 is a Lagrangian immersion of a
→ ×
surface Σ and p Σ verifies C2(p)=0, then there exists an open neighborhood U
around p such th∈at φ,ψ : U S26are diffeomorphisms onto their images, and so
→
Φ(U) is the graph of F = ψ φ 1.
−
Globally, if Φ:Σ S2−S2◦is a Lagrangianimmersion of a connected compact
→ ×
surface Σ such that C2 has no zeroes, then Σ is a sphere, Φ is an embedding and
Φ(Σ)isthegraphoftheanareapreservingdiffeomorphism(withtheoppositesign)
of the sphere S2.
Remark 2. The result proved in Proposition 4 is not true if we do not assume
the compacity of the surface, because we can construct examples of non compact
Lagrangian surfaces in S2 S2 whose associated Jacobian is constantly λ, for any
λ (0,1/2). In fact, we co×nsider the map F :S2 N,S S2 defined by
∈ −{ }→
F(x,y,z)=( ei√1−λ4λ2 tanh−1z(x+iy),z),
−
where N =(0,0,1) and S =(0,0, 1). Then, it is straightforwardto check that F
isanareapreservingdiffeomorphis−mfromS2 N,S ontoitself, andsothe graph
−{ }
8 ILDEFONSOCASTROANDFRANCISCOURBANO
of F,
−
S2 N,S S2 S2
−{ }−→ ×
(x,y,z) (x,y,z),(ei√1−λ4λ2 tanh−1z(x+iy), z) ,
7→ −
(cid:18) (cid:19)
is a Lagrangian embedding. It is an easy exercise to verify that the above La-
grangiangraph has constant associated Jacobian C =λ.
3.2. Gauss maps of certain surfaces of R4. Let Ψ :Σ R4 be an immersion
→
of an oriented surface Σ, and
Φ:Σ G+(2,4)
−→
Φ(p)=dΨ (T Σ),
p p
its Gauss map. If e ,e is an oriented orthonormal basis of dΨ (T Σ), taking
1 2 p p
{ }
into account the identification given in section 2, Φ can be written as
Φ=(φ,ψ):Σ S2 S2 Λ2R4,
→ +× − ⊂
where
1 1
φ(p)= (e e +e e ) E1(p), ψ(p)= (e e e e ) E1(p),
√2 1∧ 2 3∧ 4 ≡ + √2 1∧ 2− 3∧ 4 ≡ −
being e ,e ,e ,e an oriented orthonormal frame of R4 at Ψ(p). Looking at Φ
1 2 3 4
{ }
in Λ2R4 R6, we note that Φ(p) = φ(p) + ψ(p) = √2(e e ) while Φˆ(p) =
1 2
≡ ∧
φ(p) ψ(p)=√2(e e ).
3 4
− ∧
For any vector v T Σ, it is easy to obtain that
p
∈
dφ (v)=( σˆ(v,e ),e + σˆ(v,e ),e )E2(p)
p h 2 3i h 1 4i +
(3.1) +( σˆ(v,e ),e σˆ(v,e ),e )E3(p),
h 2 4i−h 1 3i +
dψ (v)=( σˆ(v,e ),e σˆ(v,e ),e )E2(p)
p 2 3 1 4
h i−h i −
+( σˆ(v,e ),e + σˆ(v,e ),e )E3(p),
2 4 1 3
h i h i −
where σˆ stands for the second fundamental form of the immersion Ψ. Hence, a
point p Σ verifies that dimKerdΦ > 0 if and only if there exists a non null
p
∈
vector v T Σ such that σˆ(v,w) = 0, w T Σ, i.e. the index of relative nullity
p p
∈ ∀ ∈
(see [CK]) of Ψ at p is positive.
Using thatJE2(p)=E3(p)andthe RicciequationofΨ,from(3.1)wegetthat
± ±
(3.2) dΦ (v),dΦ (w) =2 2 σˆ(v,w),Hˆ Kˆ v,w ,
p p
h i h i− h i
JdΦ (v),dΦ(cid:16)(w) = 2Rˆ (v,w,e ,e(cid:17)),
p p ⊥ 3 4
h i −
v,w T Σ, where Hˆ, Kˆ and Rˆ are the mean curvature vector, the Gauss cur-
p ⊥
∀ ∈
vatureandthe normalcurvatureoperatorrespectivelyassociatedto the immersion
Ψ. Using (3.2) we arrive at the following conclusion:
Example 2. The Gauss map Φ : Σ S2 S2 of an immersion Ψ : Σ R4 of
an oriented surface Σ is a Lagrangian→imm+e×rsio−n if and only if Ψ has flat→normal
connection and the index of relative nullity of Ψ is zero. In particular, the Gauss
map of any immersion Ψ : Σ S3 R4 of an oriented surface Σ into the 3-
→ ⊂
dimensional unit sphere is a Lagrangian immersion.
MINIMAL LAGRANGIAN SURFACES IN S2 S2 9
×
Remark 3. If Ψ is a minimal immersion of an oriented surface (with flat normal
connectionandnegativeGausscurvature),from(3.1)wegetthatbothcomponents
φ and ψ of its Gauss map Φ are conformal and (2.11) says that C2 1/4. In this
≡
case, Φ(Σ) must be congruent to an open set of M according to Proposition 2.
0
If we consider the totally geodesic S2 S3 R4, then it is an exercise to check
⊂ ⊂
that its Gauss map is also the totally geodesic Lagrangian surface M defined in
0
(2.13).
It is also easy to obtain that the Gauss map of the Clifford torus (z,w) S3
{ ∈ ⊂
C2/ z = w =1/√2 is given by
| | | | }
(z,w) S1(1/√2) S1(1/√2) 2((0, zw),(0,z¯w)) S2 S2,
∈ × 7→ − ∈ ×
where S2 = (x,z) R C/x2+ z 2 = 1 . We note it is a two fold covering of
{ ∈ × | | }
the totally geodesic Lagrangiantorus T given in (2.10).
4. Minimal Lagrangian surfaces
LetΦ=(φ,ψ):Σ S2 S2beaLagrangianimmersionofasurfaceΣ. If e ,e
1 2
→ × { }
is an orthonormal frame in Σ, then e ,e ,Je ,Je is an orthonormal frame in
1 2 1 2
Φ T(S2 S2)and,using(2.3)and(2.{4),wededuceth}atthe meancurvaturevector
∗
H¯ of Σ i×nto R6 is given by
1
(4.1) H¯ =H +H˜ =H Φ.
− 2
We consider a local isothermal parameter z =x+iy on Σ, in such a way that
(4.2) Φ ,Φ = φ ,φ + ψ ,ψ =0,
z z z z z z
h i h i h i
Φ 2 = φ 2+ ψ 2 =e2u/2,
z z z
| | | | | |
where the derivatives respect to z and z¯ are given by ∂ = 1 ∂ i ∂ , ∂ =
z 2 ∂x − ∂y z¯
1 ∂ +i ∂ . Taking into account that Φ ,JΦ = Φ ,JΦ (cid:16), the Gau(cid:17)ss equa-
2 ∂x ∂y h zz z¯i h zz¯ zi
tio(cid:16)n of Φ an(cid:17)d (4.1) imply that
e2u 1
(4.3) Φ = H Φ ,
zz¯
2 − 2
(cid:18) (cid:19)
1
Φ =2u Φ + H,JΦ JΦ +2e 2u Φ ,JΦ JΦ Φ ,Φˆ Φˆ,
zz z z z z − zz z z¯ z z
h i h i − 2h i
where Φˆ =(φ, ψ).
−
Also, from (2.9) and (4.2) we obtain that
(4.4) Φ 2 =2φ 2 =2ψ 2 =e2u/2.
z z z
| | | | | |
Finally, using (2.5) we can write
(4.5) JΦ ,Φˆ = ie2uC
z¯ z
h i −
and deduce that
(4.6) Φˆz =2e−2u Φz,Φˆz Φz¯ 2iCJΦz.
h i −
This yields
(4.7) e4u(1 4C2)=4 Φ ,Φˆ 2 =16 φ ,φ 2 =16 ψ ,ψ 2.
z z z z z z
− |h i| |h i| |h i|
10 ILDEFONSOCASTROANDFRANCISCOURBANO
4.1. Lagrangian surfaces in S2 S2 with parallel mean curvature vector.
We first classify the compact Lagra×ngiansurfaces of S2 S2 with non null parallel
×
mean curvature vector.
Theorem 1. Let Φ : Σ S2 S2 be a Lagrangian immersion with non null
→ ×
parallel mean curvature vector of a compact surface Σ. Then Φ is congruent to a
finite covering of one of the embedded tori
T = (x,y) S2 S2/x =a,y =b ,
a,b 1 1
{ ∈ × }
with a,b [0,1), a2+b2 >0.
∈
Proof. SincethemeancurvaturevectorH isanonnullparallelvectorfieldandthe
immersionis Lagrangian,JH is also a non trivial parallelvector field on Σ. Hence
the surface Σ is flat. Using (2.7), this implies that σ 2 =4(H 2+C2). Therefore,
| | | |
taking the two-fold oriented covering of Σ if necessary, Σ is a torus.
Using (4.3), we can check that the 1-differential Υ on Σ defined by
Υ(z)= H,JΦ dz,
z
h i⊗
is holomorphic. Since Υ does not vanish because H is non null, we can normalize
it by Υ = (1/2)(dz), i.e. 2 H,JΦ = 1. Derivating this equality and using again
z
h i
(4.3), we have that
0= H,JΦ = H,2u JΦ =u .
z z z z z
h i h i
This means that u is constant, say µ R, and so H 2 =4e 2µ H,JΦ 2 =e 2µ.
− z¯ −
∈ | | h i
We define now a 4-differential on Σ by
Ξ(z)= 8 H,JΦ Φ ,JΦ + Φ ,Φˆ 2 (dz)4
z zz z z z
h ih i h i ⊗
=(cid:16) 4 Φ ,JΦ + Φ ,Φˆ 2 (dz(cid:17))4.
zz z z z
h i h i ⊗
(cid:16) (cid:17)
Using (4.3), (4.6) and the fact that u is constant, it is easy to prove that Ξ is also
holomorphic. Since Σ is a torus, Ξ can be written as Ξ = λeiθ(dz)4, with λ 0
and θ R. ≥
∈
Underthese conditions,wearegoingtocompute thegradientandthe Laplacian
of C. From (4.5), (4.3) and (4.6), we obtain that
e 2µ
iCz = − Φz,Φˆz 2e−4µ Φzz,JΦz Φz¯,Φˆz¯ .
2 h i− h ih i
Since C 2 =4e 2µ C 2, using (4.2) and (4.7), we obtain that
− z
|∇ | | |
1 4C2
C 2 = − e−2µ+2C2 8e−8µ ( Φzz,JΦz Φz¯,Φˆz¯ 2),
|∇ | 2 − ℜ h ih i
where denotes real part.(cid:0)But as Ξ=λ(cid:1)eiθ(dz)4 we have that
ℜ
λ2 =e4µ+4e6µC2+ e8µ 1 4C2 2+8 ( Φ ,JΦ Φ ,Φˆ 2).
zz z z¯ z¯
16 − ℜ h ih i
The last two equations allow to ge(cid:0)t that (cid:1)
8e 2µ(1+4C2)+(1 4C2)(1+12C2)
(4.8) C 2=e−4µ(1 λ2e−4µ)+ − − .
|∇ | − 16
On the other hand, derivating C with respect to z¯ and using (4.3) and (4.6), we
z
obtain that
e2µ
C = C 1+ +e2µC2 .
zz¯
− 4
(cid:18) (cid:19)
