Parallel mean curvature surfaces in four-dimensional homogeneous spaces

0.24 MB·

by  José M. Manzano

Parallel mean curvature surfaces in four-dimensional
homogeneous spaces
JoséM.Manzano,Francisco Torralbo,andJoeriVanderVeken
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Abstract. We survey different classification results for surfaces with parallel
2
mean curvature immersed into some Riemannian homogeneous four-manifolds,
r
a including real and complex space forms and product spaces. We provide a com-
M mon framework for this problem, with special attention to the existence of holo-
morphicquadraticdifferentialsonsuchsurfaces. Thecaseofsphereswithparallel

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meancurvatureisalsoexplainedindetail,aswellasthestate-of-the-artadvances
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inthegeneralproblem.

]
G Keywordsandphrases. Parallelmeancurvature,constantmeancurvature,holo-
D morphic quadraticdifferentials,Thurstongeometry.
. MSC2010. Primary53C42;Secondary53C30.
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t
a
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1 Introduction
[

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Surface Theory in three-dimensional manifolds is a classical topic in Differential Geometry.
v
0 Although the most extensive investigation has been carried out in ambient three-manifolds
4 withconstantcurvature,the so-calledspaceforms,therehasbeenagrowing interestinconsid-
7
eringthebroaderfamilyofhomogeneousthree-manifolds. Amongthedifferentgeometrically
3
distinguishedfamiliesofsurfaces,wewillfocusonthosewhichhaveconstantmeancurvature
0
. (cmc in the sequel). When the codimension is bigger than one, a natural generalization of
1
cmcsurfacesarethosewhosemeancurvaturevectorisnotconstantbutparallelinthenormal
0
7 bundle. These surfaces are called parallel mean curvature surfaces (pmc from now on, see
1 Definition 2.1),andenjoysomeofthe propertiesofcmcsurfacesincodimension one.
:
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The aim of this work is to gather some results on the classification of pmc surfaces when
i
X the codimension is two and the ambient space is homogeneous, sketching a parallelism with
r cmc surfaces in homogeneous three-manifolds. The connection between these two theories
a
principally comes from the fact that cmc surfaces in totally umbilical cmc hypersurfacesof a
Support: Spanish MEC-Feder Research Project MTM2014-52368-P (first and second authors), EPSRC Grant No.
EP/M024512/1(firstauthor);BelgianInteruniversityAttractionPoleP07/18“Dynamics,GeometryandStatistical
Physics" (second and third authors); KU Leuven Research Fund project 3E160361 “Lagrangian and calibrated
submanifolds"(thirdauthor).TheauthorswouldliketothankProf.K.Kenmotsuforhisvaluablecomments.The
secondauthorwouldalsoliketothanktheGeometrySectionoftheDepartmentofMathematicsatKULeuvenfor
theirkindnessduringhisstayatKULeuven.
1
four-manifold become pmcsurfaces(cf. Proposition 3.1). When the four-manifold is homoge-
neous typically such a hypersurface is also homogeneous. Nonetheless, there could be pmc
surfaces not factoring through a hypersurface in this sense, as we will discuss below. The
interested reader can refer to [DHM09] and [MP12] for an introduction to cmc surfaces in
homogeneous three-manifolds. Another approach that covers both cmc and pmc surfaces as
criticalpointsofextendedareafunctionalscanbefoundin[Sal10].
In the seventies, Ferus [Fer71] proved that an immersed pmc sphere in a space form is a
round sphere (cf. Theorem 4.1), and afterwardsChen [Che73] and Yau [Yau74] classified pmc
surfaces in space forms, showing that they are cmc surfacesin three-dimensional totally um-
bilical hypersurfaces (cf. Theorem 5.1). It is also important to mention the contribution of
Hoffman [Hof73], who classified pmc surfaces of R4 and S4 in terms of analytical functions
assumingtheirGausscurvaturedoesnotchangesign.
Almost thirty years later, Kenmotsu and Zhou [KZ00] undertook the classification of pmc
surfacesin the complex spaceforms CP2 and CH2, basedon a resultof Ogata [Oga95]. How-
ever, soon thereafter Hirakawa [Hir06] pointed out a mistake in Ogata’s equation, but gave
a classification of the pmc spheres in CP2 and CH2. The mistake was corrected in [KO15]
but the classification was still incomplete. Finally, Kenmotsu has recently published a correc-
tion [Ken16] that closes the classification problem. The complete classification then follows
fromboth[Hir06]and[Ken16].
The classification of pmc surfaces in four-dimensional manifolds has also been treated in
M3(c)×R,whereMn(c)denotesthen-dimensionalspaceformofconstantsectionalcurvature
c. On the one hand, de Lira and Vitório [dLV08] classified the pmc spheres (cf. Theorem 4.4).
Ontheotherhand,Alencar,doCarmoandTribuzy[ACT10]provedreductionofcodimension
forpmcsurfacesinMn(c)×R(cf.Theorem5.2)alsoclassifyingpmcspheres(cf.Theorem4.5).
Mendonça and Tojeiro [MT14] improved Alencar, doCarmo and Tribuzy’sresultunder some
additionalconditions (seeSection5.1).
Afewyearsago,thesecondauthorandUrbano[TU12]classifiedthepmcspheresintheprod-
uctfour-manifoldsS2×S2 andH2×H2,aswellasalargefamilyofpmcsurfacesthatsatisfy
an extra condition on the extrinsic normal curvature (cf. Theorem 5.4). Fetcu and Rosenberg
alsotackled the problemin other ambient manifoldsobtaining severalpartialresults, namely,
inS3×RandH3×R[FR12],inMn(c)×R[FR13],inCPn×RandCHn×R[FR14a]andalso
inSasakianspaceforms[FR14b],includingtheHeisenbergspaceofanyodddimension.
An interesting family where to study the classification problem for pmc surfaces in is that of
the four-dimensionalThurston geometries, i.e., homogeneous four-manifoldswhose isometry
group acts transitively and effectively on them, and the stabilizer subgroup at each point is
compact. Usually the isometry group is required to be maximal in the sense that it cannot
be enlarged to another subgroup. Under these assumptions, there are 19 types of Thurston
geometriesindimension4,listedinTable1below. Wewillemphasizetheproductgeometries,
whichmightbethefirstspaceswherepmcsurfaceshould beunderstood:
• The product spaces M3(c)×R (see Sections 4.4 and 5.2). The classification of the pmc
sphereswasdone by deLiraand Vitório [dLV08], butthe generalclassification remains
open.
2
• TheproductspacesM2(c )×M2(c ). Theclassificationofspheresisknownwhen c =
1 2 1
c (see Sections 4.3 and 5.4), but the general case remains still open, although there are
2
some partialresults(seeSection4.5).
• The product spaces Nil ×R, Sl (R)×R and Sol3 ×R (the latter is included in the
3 2
familySol4 inTable1).
m,n e
Dropping the condition on the maximality of the isometry group, a simply connected homo-
geneous four-dimensional product manifold is either of the form M2(c )×M2(c ) or G×R,
1 2
where G isaLiegroupendowedwithaleft-invariantmetric(see[MP12]). Inthelatterfamily,
it is worth highligthing the family E(κ,τ)×R where E(κ,τ), κ−4τ2 6= 0, denotes the two-
parameterfamilyofsimplyconnectedthree-manifoldswithisometrygroupofdimensionfour
(see[VdV08],[DHM09]andthereferencestherein).
Theexistenceofholomorphic quadraticdifferentialsforpmcsurfaceshasbeencentralintheir
classification. Note that cmc surfaces in E(κ,τ)-spaces admit a holomorphic quadratic dif-
ferential called the Abresch-Rosenberg differential [AR05]. This fact plays a key role in the
definition of holomorphic quadratic differentialsfor pmc surfaces in H3×R and S3×R (see
Section5.2).
Geometry Isotropy dim(Iso) Kähler
S4,R4,H4 SO 10 No, exceptR4
4
CP2,CH2 U 9 Yes
2
S3×R,H3×R SO 7 No
3
S2 × S2, H2 × H2, S2 × R2, SO ×SO 6 Yes
2 2
S2×H2,H2×R2
Sl (R)×R,Nil ×R,Sol4 SO 5 No
2 3 0 2
Fe4 (S1) 5 Yes
1,2
Nil ,Sol4 ,Sol4 {1} 4 No
4 m,n 1
Table1:List of Thurston four-dimensional geometries, their isotropy group (cf. [Wal86, §1]
and [Mai98]), the dimension of their isometry group, and whether or not they admit
aKählerstructurecompatiblewiththegeometricstructure(cf.[Wal86,Theorem1.1]).
The spacesSl (R)×R, Nil ×R, Sol4, andSol4 arenotKähler butdoadmitcomplex
2 3 0 1
structures. Here, (S1) denotes the image of the unit circle S1 ⊂ C in U by z 7→
e m,n 2
(zm,zn).
It is worth mentioning that pmc surfaceshave been also studied in pseudo-Riemannian man-
ifolds. A classification wasachievedfor non-degenerate pmcsurfacesin the four-dimensional
Lorentzianspaceforms[CV09]. Itturnsoutthat,asintheRiemanniancase,allpmcsurfaceslie
in three-dimensional submanifolds. This classification was afterwards extended to any codi-
mensionandanysignatureofthemetric(see[Che09]forthespacelikecaseand[Che10,FH10]
forthetimelike case). Inthe sequelwewillrestrictourselvestotheRiemanniancase.
3
2 Definitions and first properties
Let M be an n-dimensional orientable Riemannian manifold with metric h,i and Levi-Civita
connection ∇, and let φ : Σ → M be an isometric immersion of an orientable Riemannian
surface Σ. The tangent space T Σ will be identified with dφ(T Σ) ⊂ T M in the sequel, so
p p φ(p)
themetriconΣwillalsobedenotedbyh, isincetheimmersionisisometric. ThereforeT M
φ(p)
admitsanorthogonaldecomposition T M= T Σ⊕T⊥Σ,whereT⊥Σistheso-callednormal
φ(p) p p
bundle of the immersion. This leads to considering the space X(Σ) of (tangent) vector fields,
i.e., smooth sections of TΣ, and the space X⊥(Σ) of normal vector fields, i.e., smooth sections
of T⊥Σ. We will denote by u⊤ ∈ T Σ and u⊥ ∈ T⊥Σ the components of a vector u ∈ T M
p p φ(p)
withrespecttothisdecomposition.
Given a normal vector field η ∈ X⊥(Σ), we can define the shape operator associated with η
as the self-adjoint endomorphism A : X(Σ) → X(Σ) given by A (X) = −(∇ η)⊤. Then the
η η X
second fundamental form σ : X(M)×X(M) → X⊥(Σ) satisfies hσ(X,Y),ηi = hA (X),Yi for
η
all X,Y ∈ X(Σ) and η ∈ X⊥(Σ). The mean curvature vector H of the immersion at p ∈ Σ is
definedas H(p)= 1(σ(e ,e )+σ(e ,e )),where{e ,e } isanorthonormal basisof T Σ.
2 1 1 2 2 1 2 p
ThenormalbundleT⊥Σcanalsobeendowedwithaconnection∇⊥ :X(Σ)×X⊥(Σ) →X⊥(Σ)
defined as ∇⊥η = (∇ η)⊥ for all X ∈ X(Σ) and η ∈ X⊥(Σ). This connection is called the
X X
normal connection, and gives rise to a curvature tensor R⊥ : X(Σ)×X(Σ)×X⊥(Σ) → X⊥(Σ),
givenby
R⊥(X,Y)η = ∇⊥∇⊥η−∇⊥∇⊥η−∇⊥ η, X,Y ∈X(Σ), η ∈X⊥(Σ). (2.1)
X Y Y X [X,Y]
Definition 2.1 (Parallel mean curvature immersion). An isometric immersion φ : Σ → M is
said to have parallel mean curvature (pmc for short) if its mean curvature vector H ∈ X⊥(Σ)
isparallelinthe normalbundle,i.e.,∇⊥H =0,butnotidenticallyzero.
Remark 2.1. The minimal case H = 0 has been excluded from Definition 2.1 due to several
reasons that will become clear after Lemma 2.1. Essentially, it is not possible to define a
natural orthonormal frame in the normal bundle if H = 0, which is crucial for some of the
argumentsbelow.
Although several results in higher codimension will be mentioned hereinafter, let us assume
now that the codimension is 2, where pmc surfaces enjoy additional properties. In the first
place,wewillfurnishthenormalbundlewithanaturalorientationprovidedthatboth Mand
Σ areoriented,anddefineanotionof curvatureinthenormalbundle.
Definition2.2. A basis {η,ν} in T⊥Σ is said to be positivelyoriented if and only if {e ,e ,η,ν}
p 1 2
ispositivelyorientedin T M whenever{e ,e }isapositivelyorientedbasisof T Σ.
φ(p) 1 2 p
Thenormalcurvatureof φ isthesmooth function K⊥ ∈ C∞(Σ)definedby
K⊥(p)=hR⊥(e ,e )e ,e i, (2.2)
1 2 3 4
where{e ,e ,e ,e }isanorthonormalbasisofT Msuchthat{e ,e }and{e ,e }arepositively
1 2 3 4 p 1 2 3 4
orientedbasesin T Σ and T⊥Σ,respectively.
p p
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Lemma 2.1. Let φ : Σ → M be a pmc immersion. Then the mean curvature vector H has constant
length(inparticular, H nevervanishes). Ifadditionallythecodimensionis2,then:
(i) ThereexistsauniqueparallelnormalfieldH ∈X⊥(Σ)suchthattheglobalframe{H/|H|,H/|H|}
ispositivelyorientedandorthonormalinT⊥Σ.
e e
(ii) K⊥ isidenticallyzero,i.e.,thenormalbundleisflat.
(iii) TheRicciequationfortheRiemanncurvaturetensor Rof Mreads
hR(X,Y)H,ηi =h[A ,A ]X,Yi, X,Y ∈X(Σ), η ∈X⊥(Σ). (2.3)
η H
Remark2.2. Flatnessofthenormalbundleofapmcsurfaceisatypicalpropertyincodimension
2. If the codimension is bigger than 2, it is possible to define likewise the normal sectional
curvatureofthenormalbundle,butitdoesnotnecessarilyvanishforpmcsurfaces.
Proof. Since H isparallelinthenormalbundlewehave
X(|H|2)=2h∇ H,Hi=2h∇⊥H,Hi=0,
X X
for all X ∈ X(Σ), so |H| is constant on Σ. As for (i), the normal bundle is orientable in the
senseofDefinition2.2,sowecandefinearotation R ofangle π/2in T⊥Σsuchthat{η,R η}
p p p
is positively oriented for all η ∈ T⊥Σ. This rotation leaves the normal bundle of Σ invariant,
p
andshould notbeconfused withapossible complexstructureon M.
Hence H = −RH is such that {H/|H|,H/|H|} is a positively oriented global orthonormal
frame of the normal bundle. Moreover, H is also parallel since it has constant length and
e e
is orthogonal to the parallel vector field H. Given a positively oriented orthonormal frame
e
{e ,e } in TΣ, we can consider e = H/|H| and e = H/|H|, so Equation (2.1) and the fact
1 2 3 4
that H isparallelyield
e
e |H|R⊥(e ,e )e = ∇⊥∇⊥H−∇⊥∇⊥H−∇⊥ H =0,
1 2 3 e1 e2 e2 e1 [e1,e2] (2.4)
|H|R⊥(e ,e )e = ∇⊥∇⊥He−∇⊥∇⊥He−∇⊥ He =0.
1 2 4 e1 e2 e2 e1 [e1,e2]
From (2.2) and the first equation in (2.4), we get that K⊥ ≡ 0, so (ii) is proved. Finally,
given X,Y ∈ X(Σ) and ξ,η ∈ X⊥(Σ), the Ricci equation reads R(X,Y,ξ,η) = R⊥(X,Y,ξ,η)−
h[A ,A ]X,Yi,so(iii)isaconsequenceoftakingξ = H intheRicciequationandofthesecond
ξ η
identityin(2.4).
3 The relation between cmc and pmc surfaces
Parallel mean curvature surfaces are often considered the natural generalization to higher
codimensionofcmcsurfacesinthree-manifolds,soaleadingideainthestudyofpmcsurfaces
in four-manifolds is to reduce the codimension and rely on results for cmc surfaces. Our
first approach to this idea will consist in finding natural assumptions on a hypersurface N
of a four-manifold M guaranteeing that any cmc surface immersed in N has parallel mean
curvaturevectorin M. Thisisevidentif Nistotallygeodesic,butthisconditioncanberelaxed
asthefollowing resultensures.
5
Proposition 3.1. Let N be a totally umbilical cmc hypersurface of a four-manifold M. Then every
cmcsurfaceimmersedin N iseitherpmcorminimalin M.
Proof. Let φ : Σ → N be a cmc immersion with second fundamentalform σ and mean curva-
turevector H. Theimmersionφcanalsoberegardedasanimmersioninto M,soletusdenote
e
by σ and H the second fundamental form and the mean curvature vector of the immersion
e
φ : Σ → M,respectively. Wewillalsodefineσ and H asthesecondfundamentalformandthe
meancurvaturevectorof N asahypersurfaceof M,respectively.
b b
Since σ = σ+σ, taking the traceon Σ we getthat 2H = 2H+3H−σ(η,η), where η is a unit
normal vector field to φ(Σ) tangent to N. Taking into account that N is totally umbilical, i.e.,
e b e b b
σ(x,y) = hx,yiH for all x,y ∈ TN, we finally get that H = H+H. Taking the derivative of
thislastequationwithrespecttoatangentvectorfieldV ∈X(Σ)gives
b b e b
∇⊥H = ∇ (H+H) ⊥ =(∇ H)⊥+(∇ H)⊥
V V V V
=(cid:0)∇NHe+σb(V(cid:1),H) ⊥+(∇e H)⊥ b
V V
=(cid:0)(∇VNHe)⊥b+(hVe,H(cid:1)iH)⊥+(∇bVH)⊥ =(∇VH)⊥,
e e e b b
where∇N istheLevi-CivitaconnectionofNandwehavetakenintoaccountthat(∇NH)⊥ =0
V
since H hasconstantlength. We distinguishtwocases:
e
• Ief H = 0(N is a totally geodesic hypersurfaceof M), then ∇⊥H = (∇ H)⊥ = 0 and H
V V
isparallel,sowearedone.
b b
• Assume now that H 6= 0. Observe that h∇ H,Hi = 0 since H has constant length, so
V
(∇ H)⊥ isproportionaltoaunitvectorfield η, normaltoΣ, buttangentto N. Hence
V b b b b
b (∇ H)⊥ =h∇ H,ηiη = −hH,∇ ηiη =−hH,σ(V,η)iη
V V V
b =−hH,bhV,ηiHiη =b0, b b
b b
whereweusedagainthat N isatotallyumbilicalhypersurfacein M.
Remark 3.1. Under the assumptions of Proposition 3.1, the mean curvature vector H of Σ in
M is just the sum of the mean curvature vector H of Σ in N and the mean curvature vector
H of N in M, i.e., we have the orthogonal decomposition H = H+H. Hence Σ is pmcif and
e
onlyif ∇⊥H =0and Σ isnotminimalin N or N isnottotallygeodesicin M.
b e b
Let us analyse how Proposition 3.1 can be applied in different four-manifolds where totally
umbilicalsurfacesareclassifiedinordertoconstructpmcsurfaces.
1. InthespaceformsR4,S4andH4,totallyumbilicalhypersurfaceshaveconstantsectional
curvature and constant mean curvature. Hence, the pmc surfacesprovided by Proposi-
tion 3.1are cmcsurfacesin the three-dimensional space forms R3, S3 or H3 embedded
totallyumbilicallyinthe four-dimensionalspaceform.
2. There are no totally umbilical hypersurfaces in the complex space forms CP2 and CH2
[TT63]. This is one of the difficulties when trying to produce examples of pmc immer-
sions. Infact,therearenopmcspheresinCP2 orinCH2 (cf.Theorem4.2).
6
3. In S3×R and H3×R there are plenty of totally umbilical hypersurfaces since both
spacesarelocallyconformallyflat,butonlythetotallygeodesiconeshaveconstantmean
curvature [MT14]. Since totally geodesic submanifolds in a product are the product or
totallygeodesicsubmanifolds,weconcludethatsuchtotallygeodesichypersurfacesare
locallycongruenttoS3,H3,S2×R,orH2×R.
4. In a Riemannian product M2(c )×M2(c ) of two surfaces of constant Gaussian cur-
1 2
vatures c and c , with (c ,c ) 6= (0,0), the only totally umbilical hypersurfaces with
1 2 1 2
constant mean curvature are totally geodesic. Hence, they are open subsets of prod-
ucts of one surface and a geodesic in the other surface. This was proven for S2×S2
andH2×H2, wherebothfactorshavethesame curvature,in[TU12],buttheproof can
easilybeextendedtotheother cases.
5. Consider E(κ,τ)×R, the Riemannian product of a homogeneous three-space with the
Euclidean line. If κ−4τ2 = 0, the first factor has constant sectional curvature and the
classification of totally umbilical hypersurfaces with constant mean curvature has been
treated in item 3. If τ = 0 (and κ 6= 0), the first factor is either S2×R or H2×R, so
the space under consideration is either S2×R2 or H2×R2, which have been treated
in item 4. In all other cases, if was proven in [ST09, VdV08] that there are no totally
umbilical surfaces in E(κ,τ), so the only totally umbilical hypersurfacesof E(κ,τ)×R
are open parts of the slices E(κ,τ)×{t }. A more general result in G×R, where G is
0
a simply connected three-dimensional Lie group endowed with a left-invariant metric,
followsfromtheclassificationoftotallyumbilicalsurfacesin G (see[MS15]).
4 Quadratic differentials and the classification of pmc spheres
As in the theory of cmc surfaces in homogeneous Riemannian three-manifolds, the existence
of quadratic differentials that are holomorphic for pmc immersions comes in handy in some
ambientfour-manifolds. For instance,insymmetricfour-manifoldssuchas
• the spaceformsR4,S4 andH4,
• the complexhyperbolic andprojectivespacesCP2 andCH2,
• the RiemannianproductsS2×S2 andH2×H2,
it is possible to define two holomorphic quadratic differentials for pmc surfaces. It is also
hitherto possible todefine one holomophic quadraticdifferentialina fewother cases, suchas
inM3(c)×R andM2(c )×M2(c )(deLiraandVitório[dLV08]andKowalczyk[Kow11]),or
1 2
in Sasakian space forms (Rosenberg and Fetcu [FR14b]). This is instrumental, for instance, in
the classification of pmc spheres in the aforementioned spaces, for the fact that a non-trivial
holomorphic differentialvanishesoftengivespreciousinformation.
Throughout this section, we willconsider a pmcimmersion φ : Σ → M of an oriented surface
Σ into a four-manifold M with second fundamental form σ. As in the previous section, ∇
and ∇ will denote the Levi-Civita connections in Σ and M, respectively, and R will stand
for the Riemann curvature tensor of M. Also, z = x+iy will be a conformal parameter
on Σ with conformal factor e2u, giving rise to the usual basic vectors ∂ = 1(∂ −i∂ ) and
z 2 x y
∂ = 1(∂ +i∂ ).
z¯ 2 x y
7
Lemma4.1. Underthepreviousassumptions,thefollowingformulaehold:
(i) h∂ ,∂ i = 1e2u andh∂ ,∂ i=0.
z z¯ 2 z z
(ii) ∇ ∂ =∇ ∂ =0and∇ ∂ =2u ∂ .
∂z z¯ ∂z¯ z ∂z z z z
(iii) 2σ(∂ ,∂ )= e2uH.
z¯ z
(iv) hσ(∂ ,∂ ),ηi = hR(∂ ,∂ )∂ ,ηiforanyparallelnormalsectionη.
z z z¯ z¯ z z
Proof. (i)isaconsequence of z beingaconformalparameter,(ii)isadirectcomputation using
Koszul’sformulaand(iii)isstraightforwardfromthe definitionof ∂ and ∂ . Weprove(iv):
z z¯
hσ(∂ ,∂ ),ηi =h∇⊥σ(∂ ,∂ ),ηi+hσ(∂ ,∂ ),∇⊥ηi
z z z¯ ∂z¯ z z z z ∂z¯
=h(∇ σ)(∂ ,∂ )+2σ(∇ ∂ ,∂ ),ηi
∂z¯ z z ∂z¯ z z
=h(∇ σ)(∂ ,∂ )+R(∂ ,∂ )∂ ,ηi
∂z z¯ z z¯ z z
=h∇⊥σ(∂ ,∂ )−σ(∇ ∂ ,∂ )−σ(∂ ,∇ ∂ )+R(∂ ,∂ )∂ ,ηi
∂z z¯ z ∂z z¯ z z¯ ∂z z z¯ z z
=h∇⊥(1e2uH)−u e2uH+R(∂ ,∂ )∂ ,ηi =hR(∂ ,∂ )∂ ,ηi,
∂z 2 z z¯ z z z¯ z z
wherewehavetakenintoaccount(ii),(iii),thefactthatηisparallel,thedefinitionofthecovari-
antderivativeofσandtheCodazziequation(∇ σ)(Y,Z)−(∇ σ)(X,Z)=(R(X,Y)Z)⊥.
X Y
4.1 Space forms
Let M =M4(c)bethe spaceformofconstantsectionalcurvature c ∈R,anddefineinconfor-
malcoordinatesthe quadraticdifferentials
Θ(z) =hσ(∂ ,∂ ),Hidz⊗dz,
z z
(4.1)
Θ(z)=hσ(∂ ,∂ ),Hidz⊗dz,
z z
e e
where H isgiveninLemma2.1. Equation(4.1)definesgloballyΘandΘ,i.e.,theirexpressions
donotdependuponthe choiceofthe conformalparameter.
e e
Proposition 4.1. Let φ : Σ → M4(c) be a parallel mean curvature immersion of an oriented surface
Σ. ThenΘandΘdefinedby(4.1)areholomorphicquadraticdifferentials.
e
Proof. Takinginto accountthat hR(∂ ,∂ )∂ ,ηi iszerofor anynormal vectorfield η in aspace
z¯ z z
form,the statementfollowsfromLemma4.1.
Theorem 4.1 (Ferus [Fer71], see also [Hof73, Theorem 2.2]). Let φ : S → M4(c) be a pmc
immersionofa sphere S ina spaceform. Then φ(S) iscontained ina totallyumbilicalhypersurfaceof
M4(c)asaminimalsurface.
Proof. For illustration purposes, we will prove the case c = 0, that is, M4(0) = R4. Since S is
a sphere and φ is a pmc immersion, both Θ and Θ defined in (4.1) vanish. Since Θ = 0, we
obtainthat A =|H|2Id(i.e.,φ ispseudo-umbilical).
H e
8
Arguingasinthe classicalproof thatcomplete and connected totallyumbilicalsurfacesin R3
are spheres or planes, we consider the function f : S → R4 given by f(p) = H +|H |2φ(p).
p p
For anytangentvectorfieldV ∈X(S)weget
V(f) =V(H+|H|2φ) =∇ H+|H|2V =−A V+∇⊥H+|H|2V =0,
V H V
byusingthepseudo-umbilicity,andidentifyingT SwithitsimagebydφinT M4(c). Hence
p φ(p)
f isconstant a ∈R4,sothe immersionsatisfies
2
a 1
φ− = .
(cid:12) |H|2(cid:12) |H|2
(cid:12) (cid:12)
(cid:12) (cid:12)
This means that φ(S) is contained i(cid:12)n a spher(cid:12)e S3 ⊂ R4 of radius 1/|H|, which is totally
umbilical in R4 with mean curvature |H| = |H|. Thus the mean curvature H = H−H of
S as a surface of S3 is zero (observe that H and H have the same length, and H and H are
b e b
orthogonal, seeRemark3.1).
b e b
Remark 4.1. In the proof of Theorem 4.1 we have only used one of the holomorphic differ-
entials associated to the pmc immersion to get the result. Nevertheless, both holomorphic
differentialswill be neededto geta complete classification of pmcimmersions inspace forms
(cf.Theorem5.1)aswellastoclassifypmcspheresinS2×S2 andH2×H2 (cf.Theorem4.3).
Besides, [ACT10] showed that the spheres are not the only pmc surfaces in space forms for
whichthequadraticdifferentialΘvanishesidentically: thereisalsoacompletenon-flatexam-
pleinHn withnon-negativeGaussiancurvature(cf.Remark4.3).
4.2 Complex hyperbolic and projective spaces
Let us consider M = CM2(c), i.e., the complex projective or hyperbolic space of constant
holomorphic curvature c, also including C2 = CM2(0). The situation in the complex space
forms is quite similar to that of real space forms, due to the fact that Fetcu [Fet12] defined a
coupleofholomorphic quadraticdifferentialsassociatedwithpmcimmersions inCM2(c).
TheRiemanntensor ofthesespacesreads
c
R(X,Y)Z = hY,ZiX−hX,ZiY+hJY,ZiJX−hJX,ZiJY−2hX,JYiJZ , (4.2)
4n o
where J :X(M)→X(M)isthecomplexstructure,whichsatisfies:
1. J2 =−Id.
2. J isanisometry,i.e., hJX,JYi= hX,Yi.
3. J isparallel,i.e.,∇ JY = J∇ Y, being∇the Levi-CivitaconnectionofCM2(c).
X X
Proposition4.2([Fet12,Proposition 2.3andSection3.1]). Letφ : Σ →CM2(c)beapmcimmer-
sionofanorientedsufaceΣ,andletz = x+iybeaconformalparameteronΣ. Then
Θ(z)= 8|H|2hσ(∂ ,∂ ),Hi)+3chJφ ,Hi2 dz⊗dz,
z z z
(4.3)
Θ(z)= (cid:0)8i|H|2hσ(∂ ,∂ ),Hi)+3chJφ ,Hi2(cid:1) dz⊗dz,
z z z
(cid:0) (cid:1)
definetwoquadratichoelomorphicdifferentialsonΣ.e e
9
On the one hand, if c = 0, then these differentials reduce to the corresponding diffentials in
Cn ≡R2n. Ontheotherhand,theappearanceofthenewextratermhJΦ ,Hicanbemotivated
z
bythefactthatthe CodazziequationinCM2(c)isnotassimpleasinthecaseofM4(c).
Proof. TheholomorphicityfollowseasilyfromLemma4.1,fromtheexpressionoftheRiemann
tensor (4.2)andfromthefollowing equalities:
hJH,Hi =2i|H|2e−2uhJΦ ,Φ i, (JΦ )⊤ =2e−2uΦ .
z z¯ z z
e
Let us justify the first one, by showing that if {e ,e ,e ,e } is an oriented orthonormal basis,
1 2 3 4
then hJe ,e i = hJe ,e i. Let C = hJe ,e i, which satisfies C2 ≤ 1 by Cauchy-Schwarz in-
1 2 3 4 1 2
equality. If C2 = 1, then Je = ±e , so Je = ±e and we are done. If C2 < 1, let us define
1 2 3 4
e = (1−C2)−1/2(Ce +Je ) and e = (1−C2)−1/2(Je −Ce ). Then {e ,e } is an oriented
3 1 2 4 1 2 3 4
orthonormal basis spanning the same plane as {e ,e }, so they differ in a rotation of angle θ,
3 4
e e e e
anditiseasytocheckthathJe ,e i = hJe ,e i= C.
3 4 3 4
e e
Although there exist two holomorphic quadratic differentials, there is no direct proof of the
classification of the pmcspheresinCM2(c). Allthe known proofsuse the structure equations
for pmc surfaces in CM2(c) provided by Ogata [Oga95]. The proof given by Fetcu in [Fet12,
Corollary 3.2] uses the two holomorphic differentials to show that such a sphere must have
constantGausscurvature,sothe resultfollowsfrom[Hir06,Theorem1.1].
Theorem4.2([Hir06,Corollary1.2]andalso[Fet12,Corollary3.2]). Letφ : S→CM2(c)apmc
immersionofasphereS. Thenc =0andSisaroundsphereinahyperplaneofC2.
This non-existence result of pmc spheres in CH2 and CP2 contrasts with the rest of the sym-
metric spaces, where there do exist pmc spheres (cf. Theorem 4.1 and Theorem 4.3). In other
Thurston four-geometries like M3(c)×R, M2(c )×M2(c ), E(κ,τ)×R or Sol ×R, there
1 2 3
always exist pmcspheres, since H3 and the E(κ,τ)-spacesor Sol do admit cmc spheres (see
3
thecomments belowProposition 3.1).
4.3 The Riemannian products S2×S2 and H2×H2
NowletM =M2(ǫ)×M2(ǫ),whereM2(ǫ)standsforthe2-sphereS2(ǫ =1)orthehyperbolic
plane H2 (ǫ = −1). Since both S2 and H2 admit a complex structure J, we can define on M
two different (but equivalent) complex structures J = (J,J) and J = (J,−J) (see [TU12,
1 2
Section 3]). Moreover, we can define a product structure P : TM → TM as P(u,v) = (u,−v),
whichenjoysthefollowing properties:
1. P isaself-adjointlinearinvolutiveisometry ofeverytangentplaneof M.
2. J = PJ = J P
2 1 1
3. P isparallel,i.e., ∇ PY = P∇ Y forall X,Y ∈X(M).
X X
10