Extremal Sasakian Metrics on S^3-bundles over S^2

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by  Charles P. Boyer

EXTREMAL SASAKIAN METRICS ON S3-BUNDLES OVER S2
CHARLESP.BOYER
1
1
0
Abstract. InthisnoteIprovetheexistenceofextremalSasakianmetricson
2 S3-bundles over S2. These occur in a collection of open cones that I call a

n bouquet.
a
J

8
2

] 1. Introduction
G
It is well known that the S3-bundles over S2 are classified by π (SO(4)) = Z .
D 1 2
Sothereareexactlytwosuchbundles,thetrivialbundleS2×S3andonenon-trivial
.
h bundle X∞ (in Barden’snotation[Bar65]). They aredistinquished by their second
t Stiefel-Whitney class w ∈H2(M,Z ).
a 2 2
m Thereareinfinitely manySasaki-Einsteinmetrics1onS2×S3 whichbelongto a
toriccontactstructure[GMSW04b,GMSW04a,CLPP05,MS05]. These,ofcourse,
[
areallextremal,buttheyhavec (D)=0whereDisthecontactbundle. Thereisa
1
3 moreorlessobviousconstantscalarcurvatureextremalSasakianmetriconS2×S3
v
whichisnotSasaki-EinsteinandhasfirstChernclassc (D)=2(k −k )αforevery
9 1 1 2
4 pair (k1,k2) of relativelyprime positive integers. Here α is a generatorof H2(S2×
0 S3,Z), and without loss of generality we can assume that k1 >k2. We recover the
1 well known Kobayashi-Tanno homogenous Einstein metric in the case (k ,k ) =
1 2
.
2 (1,1). The Sasakian structures are constructed from the K¨ahler form ωk1,k2 =
0 k ω +k ω on S2×S2 with the product complex structure, where ω (ω ) are the
1 1 2 2 1 2
0 standardsymplecticformsonthefirst(second)factor. Themetriccorrespondingto
1 thisK¨ahlerformhasconstantscalarcurvature. OnethenformstheS1-bundleover
:
v S2×S2 whose cohomology class is [ω ]. The constant scalar curvature K¨ahler
k1,k2
i metrics lifts to a constant scalar curvature Sasakian metric on S2 ×S3 which is
X
homogeneous,hencetoric. SothereisathreedimensionalSasakiconeκasdescribed
r
a in[BGS09],andbytheopennesstheoremof[BGS08]thereisanopensetofextremal
Sasakian metrics in κ. I want to emphasize that although extremal quasi-regular
Sasakianmetrics are alwayslifts of extremal K¨ahlerianorbifold metrics, it is NOT
true that the openness theorem for extremal Sasaki metrics is obtained by simply
lifting the openness theorem [LS93, LS94] for extremal K¨ahler metrics. In this
note I shallprovethat there are many other extremal Sasakianmetrics onS2×S3
belonging to the same contact structure.
In contrast to the situation of S2×S3, until now there are no known extremal
SasakianmetricsonX∞. HereIalsoshowthatX∞admitsmanyextremalSasakian
structures belonging to the same contact structure.
InthebeginningofthisworktheauthorwaspartiallysupportedbyNSFgrantDMS-0504367.
1ForbasicmaterialconcerningSasakiangeometryIrefertoourrecentbook[BG08].
1
2 CHARLESP.BOYER
2. Review of Extremal Sasakian Metrics
Recall [BG08] that a Sasakian structure S = (ξ,η,Φ,g) on a smooth manifold
M is a contact metric structure such that ξ preserves the underlying almost CR
structure(D,J)definedbyD=kerη andJ =Φ|D,andthealmostCRstructureis
integrable. Nowwedeformthecontact1-formbyη 7→η(t)=η+tζwhereζisabasic
1-form with respect to the characteristic foliation F defined by the Reeb vector
ξ
fieldξ. Here t lies in asuitable intervalcontaining0 andsuchthatη(t)∧dη(t)6=0.
This gives rise to a family of Sasakianstructures S(t)=(ξ,η(t),Φ(t),g(t)) that we
denote by S(ξ,J¯) whereJ¯is the induced complex structureonthe quotientbundle
ν(F )=TM/L bythetrivialline bundlegeneratedbyξ.Asthenotationsuggests
ξ ξ
we always assume that S(0)=(ξ,η(0),Φ(0),g(0))=S.
Note that this deforms the contact structure (hence, the CR structure) from D
to D = kerη(t), but they are isomorphic as complex vector bundles and isotopic
t
as contact structures by Gray’s theorem (cf. [BG08]). In fact, each choice of 1-
form defines a splitting TM =L +D and an isomorphism D≈ν(F ) of complex
ξ ξ
vector bundles. The complex structure J on D defines a further splitting of the
complexifiedbundle D⊗C=D1,0+D0,1, andthe usualDolbeault type complexes
with transverse Hodge theory holds [BG08], and the same for D .
t
We assumethat M is compactofdimension2n+1with aSasakianstructure S,
and note that the associated Riemannian metric is uniquely determined by η and
Φ as
g =dη◦(Φ⊗1l)⊕η⊗η.
Following[BGS08]welets denotethescalarcurvatureofg anddefinethe“energy
g
functional” E :S(ξ,J¯)−−→R by
(1) E(g)= s2dµ ,
Z g g
M
i.e. the L2-norm of the scalar curvature. Critical points g of this functional are
called extremal Sasakian metrics. In this case we also say that the Sasakian struc-
tureS isextremal. SimilartotheK¨ahleriancase,the Euler-Lagrangeequationsfor
this functional give [BGS08]
Theorem 2.1. A Sasakian structure S ∈ S(ξ,J¯) is a critical point for the energy
functional (1) if and only if the gradient vector field ∂#s is transversely holomor-
g g
phic. In particular, Sasakian metrics with constant scalar curvature are extremal.
Here ∂# is the (1,0)-gradient vector field defined by g(∂#ϕ,·) = ∂¯ϕ. It is
g g
importanttonotethataSasakianmetricg isextremalifandonlyifthe‘transverse
metric’ gT =dη◦(Φ⊗1l) is extremalinthe K¨ahleriansensewhichfollowsfromthe
well known relation between scalar curvatures s =sT −2n where sT is the scalar
g g g
curvature of the transverse metric. It follows that
Proposition 2.2. Let S =(ξ,η,Φ,g) be a Sasakian structure on M of dimension
2n+1 and let U ⊂M be an open set such that f :U−−→Cn is a local submersion.
Then the restriction g| is an extremal Sasakian metric if and only if gT viewed
U
as a K¨ahlerian metric on f(U) is extremal. In particular, if the Sasakian structure
S is quasi-regular and (ω,Jˆ,h) is the induced orbifold K¨ahlerian structure on the
quotient Z = M/F , then g is Sasakian extremal if and only if h is K¨ahlerian
ξ
extremal. Moreover, g has constant scalar curvature if and only if h has constant
scalar curvature.
EXTREMAL SASAKIAN METRICS ON S3-BUNDLES OVER S2 3
Given a strictly pseudoconvex CR structure (D,J) of Sasaki type on a smooth
manifoldM ofdimenision2n+1,weconsiderthesetS(D,J)ofSasakianstructures
whose underlying CR structure is (D,J). The group CR(D,J) of CR transforma-
tion acts on S(D,J), and the quotient space κ(D,J) is called the Sasaki cone
[BGS08] whose dimension satisfies 1≤dimκ(D,J)≤n+1. For a strictly pseudo-
convexCRstructure(D,J)onacompactmanifoldthe groupCR(D,J)iscompact
except in the case of standard CR structure on the sphere S2n+1 [Sch95, Lee96] in
which case it is isomorphic to SU(n+1,1) [Web77]. Thus, CR(D,J) has a unique
maximal torus up to conjugacy. So, as discussed in [Boy10] it is often convenient
to consider the ‘unreduced’ Sasakicone t+ =t+(D,J) where t is the Lie algebra of
a maximal torus in the group CR(D,J) of CR transformations. This is defined by
(2) t+ ={ξ′ ∈t | η(ξ′)>0}
where η is any 1-form representing D, and is an open convex cone in t. It is
related to the Sasaki cone κ(D,J) by κ(D,J)=t+(D,J)/W where W is the Weyl
group of CR(D,J). By abuse of terminology I also refer to t+ as the Sasaki cone.
Note that with (D,J) fixed, choosing a Reeb field uniquely chooses a 1-form η
such that kerη = D, and with J also fixed Φ, hence, g are uniquely specified.
Thus,wecanthink ofthe Sasakiconet+(D,J)asconsistingofSasakianstructures
S =(ξ,η,Φ,g), so again by abuse of notation we can write S ∈t+(D,J).
Let η(t) = η + tζ where ζ is a basic 1-form that is invariant under the full
torus T. Such a 1-form can always be obtained by averaging over T. So the Lie
algebra t does not change under such a deformation, but generally the Sasaki cone
t+ associatedwith D shifts, andwe denote it by t+(t). We let ST(ξ,J¯) denote the
t
subset of S(ξ,J¯) consisting of T-invariantSasakianstructures. If S =(ξ,η,Φ,g) is
T-invariant,thenST(ξ,J¯)consistsofalldeformationsobtainedbyη 7→η(t)=η+tζ
with ζ invariant under T. We are interested in the case when the torus T has
maximal dimension and the Reeb vector field is an element of the Lie algebra t,
that is, a toric contact structure D of Reeb type.
Lemma 2.3. Let (M,D,T)be a toric contact structureof Reeb typewith Reebvec-
torfieldξ.SupposealsothatthereisanextremalrepresentativeS(t)=(ξ,η(t),Φ ,g )∈
t t
S(ξ,J¯). Then S(t)∈ST(ξ,J¯).
Proof. Since the contact structure D is toric of Reeb type, there is a compati-
ble T-invariant Sasakian structure S = (ξ,η,Φ,g) by [BG00], and suppose that
S(ξ,J¯) has an extremal representative S(t) = (ξ,η(t),Φ ,g ). By the Rukimbira
t t
ApproximationTheorem,ifnecessary,thereisaquasi-regularT-invariantSasakian
structureinvariantclosetoS,sowecanassumethatS isquasi-regular. Butthenby
Proposition 2.2 the Sasakian deformation corresponds to a K¨ahlerian deformation
on the base K¨ahler orbifold M/F . By a theorem of Calabi [Cal85] this deformed
ξ
K¨ahlerstructurehasmaximalsymmetry,andoneeasilyseesthatthecorresponding
deformed Sasakian structure S(t) also has maximal symmetry which implies that
S(t)∈ST(ξ,J¯). (cid:3)
Now let (D,J) be a strictly pseudoconvex CR structure of Sasaki type. We say
that S ∈ t+(D,J) is an extremal element2 of t+(D,J) if there exists an extremal
SasakianstructureinS(ξ,J¯). WealsosaythatS isanextremalelementofκ(D,J).
2In[BGS08]thiswascalledacanonical element,butIprefertocallitanextremalelement.
4 CHARLESP.BOYER
Sowecandefinetheextremalsete(D,J)⊂κ(D,J)asthesubsetconsistingofthose
elements that have extremal representatives (similarly for t+(D,J)).
Recall the transverse homothety (cf. [BG08]) taking a Sasakian structure S =
(ξ,η,Φ,g) to the Sasakian structure
S =(a−1ξ,aη,Φ,ag+(a2−a)η⊗η)
a
for any a∈R+. It is easy to see that
Lemma 2.4. Let S be a Sasakian structure. If S is extremal, so is S , and if S
a
has constant scalar curvature so does S . In particular, if the extremal set e(D,J)
a
is non-empty it contains an extremal ray of Sasakian structures.
Amainresultof[BGS08]saysthate(D,J)isanopensubsetofκ(D,J). Lemma
2.4 says that e(D,J) is conical in the sense that it is a union of open cones,so it is
not necessarily connected.
3. Bouquets of Sasaki Cones and Extremal Bouquets
There may be many Sasaki cones associated to a given contact structure D of
Sasaki type. They are distinguished by their complex structures J. As shown in
[Boy10]toacompatiblealmostcomplexstructureJ onacompactcontactmanifold
(M,D) one can associate a conjugacy class C (D) of maximal tori in the contac-
T
tomorphism group Con(M,D). Furthermore, almost complex structures that are
equivalent under a contactomorphism give the same conjugacy class C (D). So
T
given inequivalent complex structures J labelled by positive integers, we can as-
l
sociate unreduced Sasaki cones t+(D,J ), or the full Sasaki cones κ(D,J ). This
l l
leads to
Definition 3.1. We define a bouquet of Sasaki cones B(D) as the union
B(D)=∪l∈Aκ(D,Jl)
where A ⊂ Z+ is an ordered subset. We say the it is an N-bouquet if the cardi-
nality of A is N and denote it by B (D).
N
Clearly a 1-bouquet of Sasaki cones is just a Sasaki cone. Generally, the Sasaki
cones in B(D) can have varying dimension.
The example ofinterestto ushere hasinits foundations inthe workofKarshon
[Kar03] and Lerman [Ler03]. The general formulation is given in [Boy10]. We
begin with a simply connected symplectic orbifold (B,ω) such that ω defines an
integral class in the orbifold cohomology group H2 (B,Z). Suppose further that
orb
the class [ω] is primitive and that there are compatible almost complex structures
Jˆ on B such that (ω,Jˆ) is K¨ahler for each l in some index set A. We can asso-
l l
ciate to each such Jˆ a conjugacy class of maximal tori in the symplectomorphism
l
group Sym(B,ω). Assume that this map is injective. Now form the principal S1-
orbibundle over B associated to [ω] and assume that the total space M is smooth.
By the orbifold version of the Boothby-Wang construction we get a contact mani-
fold(M,η)satisfyingπ∗ω =dηwhereπ :M−−→Bisthenaturalorbifoldprojection
map. WecanliftacompatiblealmostcomplexstructureJˆ onBtoaD-compatible
l
almost complex structure J on D. Choosing tori T associated to Jˆ we can also
l l l
lift these to maximal tori Ξ×π−1(T ) in the contactomorphism group Con(M,D)
l
where D = kerη and Ξ is the circle group generated by the Reeb vector field ξ of
η. Suppose that there are exactly N such maximal tori in Con(M,D), then we get
EXTREMAL SASAKIAN METRICS ON S3-BUNDLES OVER S2 5
anN-bouquetB (D) ofSasakiconeson(M,D)whichintersectinthe rayofReeb
N
vector fields, ξ =a−1ξ. If we projectivize the Sasaki cones by transverse homoth-
a
etiesweobtaintheusualnotionofbouquet,namely,awedgeproduct P(κ(D,Jˆ))
l
with base point ξ. W
Suppose thatS =(ξ,η,Φ ,g ) isanextremalelementofκ(D,J )foreachl ∈A,
l l l l
that is for each l ∈A there is an extremal representative S (t)∈S(ξ,J¯). Then by
l l
the openness theorem there is a nonempty open extremal set e(D,J ) ⊂ κ(D,J )
l l
containing S =(ξ,η,Φ,g) for each l ∈A. This leads to
Definition 3.2. We define an extremal bouquet EB(D) associated with the
contact structure D to be the union ∪l∈Ae(D,Jl) if for each l ∈A the extremal set
e(D,J ) is non-empty. Moreover, it is called an extremal N-bouquet EB (D)
l N
if A has cardinality N.
One easily sees from the openness theorem of [BGS08] that EB(D) is open in
B(D). An important open problem here is to obtain a good measure of the size of
theextremalsetse(D,J ),andhence,ameasureofthesizeoftheextremalbouquet.
l
The actual size of extremal Sasakian sets is known in very few cases, namely the
standard sphere, and the Heisenberg group [BGS08, Boy09].
4. The Main Theorems
Let us now construct contact structures of Sasaki type on S2 ×S3 and X∞.
Firstconsiderthe toric symplectic manifoldS2×S2 with symplectic formω =
k1,k2
k ω +k ω onS2×S2where(k ,k )arerelativelyprimeintegerssatisfyingk ≥k ,
1 1 2 2 1 2 1 2
and ω (ω ) is the standard symplectic forms on the first (second) factor of S2, re-
1 2
spectively. Letπ :M−−→S2×S2bethecirclebundlecorrespondingtothecohomol-
ogy class [ω ]∈H2(M,Z). As stated in the introduction M is diffeomorphic to
k1,k2
S2×S3 foreachsuchpair(k ,k ). BytheBoothby-Wangconstructiononeobtains
1 2
a contact structure D on M by choosing a connection 1-form η such that
k1,k2 k1,k2
π∗ω =dη . Justin Pati [Pat09] has recently shown using contacthomology
k1,k2 k1,k2
that there are infinitely many such contact structures with the same first Chern
class that are inequivalent.
Theorem 4.1. Let M =S2×S3 with the contact structureD described above.
k1,k2
Let N = ⌈k1⌉ denote the smallest integer greater than or equal to k1. Then for
k2 k2
each pair (k ,k ) of relatively prime integers satisfying k ≥ k , there is an ex-
1 2 1 2
tremal N-bouquet EB (D ) of toric Sasakian structures associated to the con-
N k1,k2
tact structure D on S2×S3, and all the cones of the bouquet have dimension
k1,k2
three. Furthermore, the extremal Sasakian structure corresponding to the contact
1-form η has constant scalar curvature if and only if the transverse complex
k1,k2
structure is that induced by the product complex structure on the base S2×S2.
Proof. By Proposition 2.2 a Sasakian structure S = (ξ,η,Φ,g) in κ(D ,J) is
k1,k2
extremalif andonly if the induced K¨ahlerstructure (ω ,Jˆ,h) onS2×S2 is ex-
k1,k2
tremal. NowKarshon[Kar03]provedthatthereareprecisely⌈k1⌉evenHirzebruch
k2
surfacesS thatareK¨ahlerwithrespecttothe symplecticformω . Morepre-
2m k1,k2
ciselyfor eachm=0,··· ,⌈k1⌉−1 there is a diffeomorphismthat takesthe K¨ahler
k2
form S to ω . Note that m = 0 corresponds the product complex structure
2m k1,k2
CP1×CP1. Moreover, Calabi [Cal82] has shown that for every K¨ahler class [ω] of
any Hirzebruch surface S there is an extremal K¨ahler metric which is of constant
n
6 CHARLESP.BOYER
scalar curvature if and only if n = 0. Furthermore, as shown by Lerman [Ler03]
onecanliftmaximaltoriofthe symplectomorphismgroupSym(S2×S2,ω )to
k1,k2
maximal tori in the contactomorphismgroup Con(S2×S3,D ) (see also Theo-
k1,k2
rem 6.4 of [Boy10] for the more general situation). As explained above this gives
a Sasaki cone associated to each of the transverse complex structures induced by
the Hirzebruchsurfaces,andhence anN-bouquetofSasakiconeswhereN =⌈k1⌉.
k2
Furthermore, since each Sasaki cone has an extremal Sasakian structure, namely
the one with 1-form η , we can apply Theorem 7.6 of [BGS08] to obtain an
k1,k2
openset of extremalSasakianstructures in eachofthe Sasakicones. This gives an
extremal N-bouquet EB (D ) as defined above. (cid:3)
N k1,k2
There is another family of contact structures on S2 × S3 that arise as circle
bundlesoverthenon-trivialS2-bundleoverS2 whichisdiffeomorphictoCP2 blown
2
up at a point. Following [Kar03] I denote this manifold by CP . The construction
of this family also gives contact structures on the non-trivifal bundle X∞. Let E
2
be the exceptional divisor and let L be a projective line in CP that does not
intersect E. We consider the symplectic form ω˜ such that the symplectic areas
l,e f
of L and E are 2πl and 2πe, respectively, where (l,e) are relatively prime integers
satisfying l > e ≥ 1. Karshon [Kar03] proved that there are precisely ⌈ e ⌉ odd
l−e
Hirzebruchsurfaces S with m=0,··· ,⌈ e ⌉−1 that areK¨ahler with respect
2m+1 l−e
to the symplectic form ω˜ , and againthere are diffeomorphisms ψ that take the
l,e m
K¨ahler forms of S to the form ω˜ . As occured in the proof of the previous
2m+1 l,e
theoremthedifferentHirzebruchsurfacescorrespondtonon-conjugatemaximaltori
2
inthesymplectomorphismgroupSym(CP ,ω˜ ). ByProposition6.3andTheorem
l,e
6.4 of [Boy10] these can be lifted to non-conjugate tori in the contactmorphism
f
2
group Con(M,D˜ ) where M is the circle bundle over CP corresponding to the
l,e
cohomologyclass[ω ]andD˜ isthecontactstructureinducedbyaconnection1-
l,e l,e f
formη satisfyingπ∗ω =dη . Iassumethatlandearerelativelyprimepositive
l,e l,e l,e
integers. NowinordertoidentifythediffeomorphismtypeofM itisenoughbythe
Barden-Smaleclassificationofsimplyconnected5-manifoldstocompute themod2
2
reduction of the first Chern class c (D˜ ). For this we pullback c (CP ) to M and
1 l,e 1
2
use the relation [π∗ω˜l,e]=0. Now c1(CP )=2αE +αL where αE(αfL) is Poincar´e
dual of the divisor class E(L), respectively [GH78]. We find c (D˜ ) = l−2e γ
f 1 l,e
where γ is a generator of H2(M,Z) ≈ Z. It follows that M is S2×S3 if(cid:0)l is ev(cid:1)en
andX∞ ifl isodd. Then, usingCalabi[Cal82],justasinthe proofofTheorem4.1
we arrive at
2
Theorem 4.2. Let M be the circle bundle over CP with the contact structure
D˜ as described above, and let N = ⌈ e ⌉. Then for each pair (l,e) of relatively
l,e l−e f
prime integers satisfying l > e ≥ 1, there is an extremal N-bouquet EB (D˜ )
N l,e
of toric Sasakian structures associated to the contact structure D˜ , and the cones
l,e
of the bouquet all have dimension three. Furthermore, if l is even M = S2×S3,
whereas, if l is odd M = X∞ and in either case the extremal Sasakian structure
corresponding to the contact 1-form η does not have constant scalar curvature.
l,e
EXTREMAL SASAKIAN METRICS ON S3-BUNDLES OVER S2 7
Remark 4.3. Ifthepairsofintegers(k ,k )and(l,e)arenotrelativelyprime,but
1 2
(k ,k ) = (l,e)= n, similar results to Theorems 4.1 and 4.2 hold for the quotient
1 2
manifolds (S2×S3)/Zn and X∞/Zn.
5. Concluding Remarks
Recall [BG08] that a Sasakian structure S = (ξ,η,Φ,g) is said to be positive
(negative) if its basic first Chern class c (F ) can be represented by a positive
1 ξ
(negative)definite(1,1)-form. Itisnullifc (F )=0,andindefiniteotherwise. The
1 ξ
followingresultfollowsdirectlyfromTheorem8.1.14of[BG08]andProposition4.4
of [BGS08].
Lemma5.1. LetS =(ξ,η,Φ,g)beaSasakian structurewithunderlyingCRstruc-
ture (D,J). Suppose that dimκ(D,J)>1. Then the type of S is either positive or
indefinite.
In a forthcoming work [BP10] it will be seen that generally the type is not
an invariant of the Sasaki cone. Here it is easy to see from the constructions
above that the type is not an invariant of the bouquet. For example consider the
contact structure D on S2×S3. There are five Sasaki cones κ(D ,J ) where
5,1 5,1 2m
m = 0,··· ,4. We can label the corresponding extremal Sasakian structures as
S2m = (ξ,η5,1,Φ2m,g2m) where J2m = Φ2m|D5,1 and g2m = dη5,1 ◦(Φ2m ⊗1l)+
η ⊗η . TheSasakianstructureS ispositiveandhasconstantscalarcurvature;
5,1 5,1 0
whereas, the others S with m=1,2,3,4 are indefinite with non-constant scalar
2m
curvature. Similarly, the contact structure D˜5,4 on X∞ has four Sasaki cones
κ(D˜ ,J )withm=0,1,2,3. ThecorrespondingextremalSasakianstructures
4,1 2m+1
areS =(ξ,η˜ ,Φ ,g )whereJ =Φ | andg =dη˜ ◦
2m+1 4,1 2m+1 2m+1 2m+1 2m+1 D˜4,1 2m+1 5,1
(Φ ⊗1l)+η˜ ⊗η˜ . Again S is positive, and the others are indefinite, but
2m+1 4,1 4,1 1
now they are all extremal of non-constant scalar curvature.
The case (l,e) = (2,1) and m = 0 presents an interesting case even though the
SasakibouquetdegeneratestoasingleSasakicone. Herewehavec ((D˜ ,J ))=0
1 2,1 0
and M = S2 ×S3. So by Calabi the induced Sasaki structure over the complex
2
manifold CP has an extremal Sasaki metric with non-constant scalar curvature.
However,itwasshownin[MS06]thatthecontactstructureD˜ admitsanirregular
f 2,1
Sasaki-Einsteinstructure, which, of course,is extremalwith constantscalarcurva-
ture. Inthis case itwouldbe interestingto see how big the extremal sete(D˜ ,J )
l,e 0
is. This is actually a special case of a much more general result of Futaki, Ono,
and Wang [FOW09] which says that for any positive toric Sasakianstructure with
c (D) = 0 there is a deformation to a Reeb vector field in the Sasaki cone whose
1
Sasakian metric is Sasaki-Einstein. This discussion begs the questions:
Does every toric contact structure of Reeb type on S2×S3 or X∞ have a constant
scalar curvature Sasakian metric somewhere in its Sasaki cone?, or more generally
for any toric contact structure of Reeb type?
Anotherobservationcomesfromthewellknownfact[MK06]thatforeachm>0
there is a complex analytic family of even Hirzebruch surfaces S (t) satisfying
2l
S (t) ≈ S for t 6= 0 and m > l, but that S (0) ≈ S . This implies that the
2l 2l 2l 2m
moduli space of complex structures on S2×S2 is non-Hausdorff. A similar result
2
holds for the moduli space of complex structures on CP . These results imply that
themodulispaceofextremalSasakianstructureswithunderlyingcontactstructures
f
Dk1,k2 on S2×S3 or D˜l,e on X∞ is non-Hausdorff as well.
8 CHARLESP.BOYER
Note added: The first question above has been recently answered in the affir-
mative by E´veline Legendre [Leg10]. Moreover, she shows that if k > 5k the
1 2
toric contact structure D on S2×S3 has two distinct rays in the Sasaki cone
k1,k2
withconstantscalarcurvatureSasakianmetrics. SounliketheSasaki-Einsteincase
[CFO08, FOW09] constant scalar curvature rays are not unique in a Sasaki cone.
Acknowledgements
I would like to thank V. Apostolov for many discussions about extremal K¨ahler
metrics.
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DepartmentofMathematicsandStatistics,UniversityofNewMexico,Albuquerque,
NM87131.
E-mail addresses: [email protected]