The existence of Hamiltonian stationary Lagrangian tori in Kahler manifolds of any dimension

0.25 MB·English

by  Yng-Ing Lee

The existence of Hamiltonian stationary
0 Lagrangian tori in K¨ahler manifolds of any
1
0 dimension
2

n
a Yng-Ing Lee
J

1
2
Department of Mathematics and Taida Institute of Mathematical Sciences,

] National Taiwan University, Taipei 10617,Taiwan
G
National Center for Theoretical Sciences, Taipei Office
D email: [email protected]
.
h Abstract
t
a
HamiltonianstationaryLagrangiansareLagrangiansubmanifoldsthat
m
are critical points of the volume functional under Hamiltonian deforma-
[ tions. They can be considered as a generalization of special Lagrangians

or Lagrangian and minimal submanifolds. In [6], Joyce, Schoen and the
1
author show that given any compact rigid Hamiltonian stationary La-
v
1 grangian in Cn, one can always find a family of Hamiltonian stationary
6 Lagrangians of the same type in any compact symplectic manifolds with
8 a compatible metric. The advantage of thisresult is that it holds in very
3 generalclasses. Butthedisadvantageisthatwedonotknowwherethese
. examples locate and examples in this family might be far apart. In this
1
0 paper,wederivealocal condition on K¨ahlermanifolds whichensuresthe
0 existence of one family of Hamiltonian stationary Lagrangian tori near
1 a point with given frame satisfying the criterion. Butscher and Corvino
: posted a condition in n = 2 in [2]. But our condition appears to be dif-
v
ferentfromtheirs. Theconditionderivedinthispapernotonlyworksfor
i
X any dimension, but also for the Clifford torus case which is not covered
r by their condition.
a
1 Introduction
Hamiltonian stationary (or H-minimal) Lagrangians were defined and studied
byOh[9,10]inaK¨ahlermanifold(M,g). Theseobjectshavestationaryvolume
amongst Hamiltonian equivalent Lagrangians. The Euler–Lagrange equation
for a Hamiltonian stationary Lagrangian L is d α = 0, where H is the mean
∗ H
curvature vector on L, α the 1-form on L defined by α ()=ω(H, ), and d
H H ∗
· ·
the Hodge dual of the exterior derivative d.
SpecialLagrangians/Lagrangianandminimalsubmanifoldsarecriticalpoints
ofthevolumefunctionalofallvariations,andHamiltonianstationaryLagrangians
1
canbe consideredas their generalizations. HamiltonianstationaryLagrangians
are related models for incompressible elasticity theory and are closely related
to the study of special Lagrangians/ Lagrangian and minimal submanifolds.
Although there are no compact special Lagrangians in Cn, there are compact
Hamiltonian stationary Lagrangians in Cn. Oh proves in [10, Th. IV] that for
a ,...,a >0, the torus Tn in Cn given by
1 n a1,...,an
Tn = (z ,...,z ) Cn : z =a , j =1,...,n (1)
a1,...,an 1 n ∈ | j| j
(cid:8) (cid:9)
is a stable, rigid, Hamiltonian stationary Lagrangian in Cn. A Hamiltonian
stationary Lagrangian is called stable (or H-stable) if the second variational
formula of the volume functional among Hamiltonian deformations is nonnega-
tive. A Hamiltonian stationary Lagrangian in Cn is called rigid (or H-rigid) if
the Jacobi vector fields for Hamiltonian variations consist only those from the
U(n)⋉Cn actions on Cn (see [6, 2.3]). Other compact stable, rigid, Hamilto-
nian stationary Lagrangians in Cn§are given in [1].
H´elein and Romon found all Hamiltomian stationary Lagrangian tori in
C2 and CP2 via a Weierstrass-type representation [3, 4]. But there are very
few results on the existence of Hamiltonian stationary Lagrangians in general
K¨ahler manifolds. In [6], Joyce, Schoen and the author obtain families of com-
pactHamiltonianstationaryLagrangiansineverycompactsymplecticmanifold
(M,ω) with a compatible metric g. It in particular contains the case of K¨ahler
manifolds. The result is:
Theorem [6] Suppose that (M,ω) is a compact symplectic 2n-manifold, g is a
Riemannian metric on M compatible with ω, and L is a compact, Hamiltonian
rigid, Hamiltonian stationary Lagrangian in Cn. Then there exist compact,
HamiltonianstationaryLagrangians L inM whicharediffeomorphic to L,such
′
that L is contained in a small ball about some point p M, and identifying M
′
near p with Cn near 0 in Darboux geodesic normal co∈ordinates, L is a small
′
deformation of tL for small t>0. If L is also Hamiltonian stable, we can take
L to be Hamiltonian stable.
′
The method used in [6] is first to find Darboux coordinates at each point
which also admit a nice expression on the metric, and then put a scaled com-
pact Hamiltonian stationary Lagrangian from Cn in the Darboux coordinates
at each point. These submanifolds are Lagrangianin (M,ω,g), but not Hamil-
tonianstationaryyet. Onethentriesto perturbtheseapproximateexamplesin
Hamiltonian equivalence class to Hamiltonian stationary. This involves solving
a highly nonlinear equationwhose linearizedequationhas approximatekernels,
andthusitcannotbedoneingeneral. In[6],wefirstsolvetheequationperpen-
dicular to the approximate kernels for examples near any fixed point and then
show that the problem of finding Hamiltonian stationary Lagrangians, which
arecriticalpointsofthevolumefunctionalonaninfinitedimensionalspace,can
bereducedtofindingcriticalpointsofasmoothfunctiononafinitedimensional
compactspacewhenthemodelfromCn isinadditionHamiltonianrigidandM
iscompact. Theexistencewillfollowfromthesimplefactthateverycontinuous
function has critical points on a compact set.
2
The advantage of the above argument is that it only requires compactness
and works in very general classes. But the disadvantage is that we do not
know where the Hamiltonian stationary Lagrangian locates in (M,ω,g). As
a consequence, the examples obtained at each scaled size t may be far apart
for different t. In this paper, we take a different approach in the second step
and resolve this deficit when M is a K¨ahler manifold and L = Tn . More
a1,...,an
precisely, we show that
Theorem4.4Supposethat (M,ω,g)isann-dimensionalK¨ahlermanifold, and
write U for the U(n) frame bundle of M. The subgroup of diagonal matrices
Tn U(n) acts on U. For any given a > 0, i = 1,...,n, define F :
U/T⊂n →R by Fa1,...,an(p,[υ])= ni=1a2iiRi¯ii¯i(p), where p∈M, υ ∈Ua(1n,)..,.,aannd
theholomorphicsectionalcurvaturPeRi¯ii¯i(p)iscomputedw.r.t. theunitaryframe
υ at p whose value is clearly independent of the representative υ of [υ]. Assume
that (p ,[υ ]) U/Tn is a non-degenerate critical point of F , then for t
0 0 ∈ a1,...,an
small there exist asmooth family (p(t),[υ(t)]) U/Tn satisfying (p(0),[υ(0)])=
∈
(p ,[υ ]) and a smooth family of embedded Hamiltomian stationary Lagrangian
0 0
tori with radii (ta ,...,ta ) center at p(t) which are invariant under Tn action
1 n
and posited w.r.t any representative of [υ(t)]. Moreover, the distance between
(p(t),[υ(t)]) and (p ,[υ ]) in U/Tn is bounded by ct2. The family of embedded
0 0
Hamiltomian stationary Lagrangian tori do not intersect each other when t is
small.
The proof of the theorem is along the same line as in [6] with the following
differences:
OnaK¨ahlermanifold,weobtainDarbouxcoordinateswithbetterexpres-
•
sionsonthe metric. AndwhenL=Tn ,wecancomputethe leading
a1,...,an
terms in related estimates in explicit forms.
In the last step, instead of using compactness to show the existence, we
•
analyze directly the conditions we need to perturb approximate examples
toHamiltonianstationary. Thisisdonebyderivingexplicitexpressionsup
tosomeordersinallrelatedestimates. Becausewedonotusecompactness
condition, the result also hold for noncompact K¨ahler manifolds.
Our result is an analogue to the constant mean curvature (CMC) hyper-
surface case in a Riemannian manifold M. Ye in [12] showed that near a non-
degenerate critical point p of the scalar curvature function on M, there exist
CMCspherefoliationnearp. Theproblemoffindingacorrespondingcondition
for Hamiltomian stationary Lagrangian tori on a K¨ahler manifold is proposed
bySchoen,andisthestartingpointofourprojectinthisdirectionincluding[6].
ButscherandCorvinoproposedadifferentconditionin[2]forn=2,whichisthe
non-degeneratecriticalpointofthefunctionG (p,υ)=a2RC (p)+a2RC (p)
a1,a2 1 1¯1 2 2¯2
on U/T2. Here RC and RC are the complex Ricci curvature. Note that G
1¯1 2¯2 a1,a2
willbeamultipleofthe scalarcurvaturewhena =a . Itisindependentofthe
1 2
frame, and thus there won’t be any non-degenerate critical point of G in
a1,a2
this case. In contrast to that, our condition not only works for any dimension,
3
but also cover the Clifford torus (i.e., with same radii) case. Because the di-
mensiondoes not match, the family ofHamiltomian stationaryLagrangiantori
with radii (ta ,...,ta ) won’t form a foliation.
1 n
This paperisorganizedasfollows. In 2wegivebasicdefinitions andderive
§
new Darboux coordinates which will play an essential role in the paper. Some
importantandinvolvingestimatesaregivenin 3. Section4consistofthesetup
§
fortheperturbationandtheproofofthemaintheorem. Adifferentproofwhich
is more close to our approach in [6] is presented in the last section. This more
geometrical simple proof also give another justification for the computation in
3. The first proof has its own interests which demonstrate the general scheme
§
of the perturbation method. So we present both proofs in the paper.
Acknowledgements: I benefit a lot from the joint project with Joyce and
Schoenin[6],andthesetupofthispaperverymuchfollowsthatin[6]. Iwould
liketoexpressmyspecialgratitudetobothofthem. Brendle’scommentsinmy
talk in Columbia university remind me to revisit one of my earlier approaches
andleads to the differentproofinthe lastsection. I amindebted to him for his
enlightening comments, and also to Joyce for his many helpful comments in a
earlier version of the paper.
2 Notation and Darboux coordinates
2.1 Lagrangian and Hamiltonian stationary
We will assume(M,ω,g)to be K¨ahlerthroughthis paper,andrefer to 2 in[6]
§
for more detailed discussions on the background material.
Definition 2.1. A submanifold L in (M,ω) is called Lagrangian if dimL =
n= 1dimM and ω 0. It follows that the image of the tangent bundle TL
2 |L ≡
under the complex structure J is equal to the normal bundle T L.
⊥
LetF :M RbeasmoothfunctiononM. TheHamiltonianvectorfield v
F
→
ofF istheuniquevectorfieldsatisfyingv ω =dF. TheLiederivativesatisfies
F
·
ω = v dω +d(v ω) = 0, so the trajectory of v gives a 1-parameter
LfavmFily of dFiff·eomorphisFm·s Exp(sv ) : M M for s RFwhich preserve ω. It
F
→ ∈
is called the Hamiltonian flow of F. If L is a compact Lagrangian in M then
Exp(sv )L is also a compact Lagrangianin M.
F
Definition 2.2. A compact Lagrangian submanifold L in (M,ω,g) is called
Hamiltonian stationary, or H-minimal, if it is a critical point of the volume
functionalamongHamiltoniandeformations. Thatis,LisHamiltonianstation-
ary if
d Vol Exp(sv )L =0 (2)
ds g F s=0
(cid:0) (cid:1)(cid:12)
for all smooth F : M R. By Oh [10, Th. I(cid:12)], (2) is equivalent to the Euler–
→
Lagrange equation
d α =0, (3)
∗ H
4
whereH isthemeancurvaturevectorofL,andα =(H ω) istheassociated
H L
· |
1-form of H on L, and d is the Hodge dual of the exterior derivative d on L,
∗
computed using the metric h=g .
L
|
When (M,ω,g) is a Calabi-Yau manifold, one can choose a holomorphic
(n,0)-form Ω on M with Ω=0, normalized so that
∇
ωn/n!=( 1)n(n 1)/2(i/2)nΩ Ω¯.
−
− ∧
If L is an oriented Lagrangian in M, then Ω eiθdV , where dV is the
L L L
| ≡
induced volume form from the metric g. It defines Lagrangian angle θ : L
R/2πZ on L. The submanifold L is called special Lagrangian if θ is constan→t.
On a Hamiltonian stationary Lagrangian, the Lagrangian angle θ is harmonic.
Ifmoreover,the imageofθ liesinR(andhereLiscompact),thentheHamilto-
nianstationaryLagrangianLisindeedspecialLagrangiansinceeveryharmonic
function on a compact manifold must be constant.
At a Hamiltonian stationary LagrangianL, one can compute to find that
d2 Vol Exp(sv )L = f,f , (4)
ds2 g F s=0 L L2(L)
(cid:0) (cid:1)(cid:12) (cid:10) (cid:11)
(cid:12)
where f =F and
L
|
Lf =∆2f +d∗αRic⊥(J∇f)−2d∗αB(JH,∇f)−JH(JH(f)). (5)
In (5), ∆f = d df is the positive Laplacian on L, B(, ) is the second fun-
∗
· ·
damental form on L, and Ric⊥(v) is a normal vector field along L defined by
Ric(v,w) = Ric⊥(v),w for any normal vector w. Note also that by the La-
h i
grangiancondition JH is tangent to L.
Givenasmoothfunctionf onaLagrangianL,wecanextendittoasmooth
function F on M and consider L = Exp(sv )L whose mean curvature vector
s F
isdenotedbyH . Onecanderivethe linearizedoperatorof d α = d α
s − ∗ Hf − ∗ H1
and obtain that
d
(d α ) = f. (6)
− ds ∗ Hs s=0 L
(cid:12)
Here L does not need to be Hamiltonian(cid:12)stationary.
Oh proves in [10, Th. IV] that the torus Tn in Cn given by (1) is
a1,...,an
Hamiltonianstationarywith(4)nonnegativedefinite (Hamiltonianstable),and
Ker at Tn consist of functions of the following form
L a1,...,an
Q(z ,...,z )=a+ n (b z +¯b z¯ )+ n c z z¯
1 n j=1 j j j j j=k jk j k
P P 6
restricted on Tn , where a R, b , c C, and c = c¯ (Hamiltonian
a1,...,an ∈ j jk ∈ jk kj
rigid, see [6]). If we write Tn in polar coordinates
a1,...,an
Tan1,...,an = (a1e√−1θ1,...,ane√−1θn)∈Cn :θi ∈[0,2π),i=1,...,n , (7)
(cid:8) (cid:9)
then Ker is spanned by
L
1, cosθ , sinθ , cos(θ θ ), sin(θ θ ) (8)
i i i j i j
− −
for i, j =1,...,n and i=j [10].
6
5
2.2 New Darboux coordinates
The convention for curvature operator used in this paper is
R(X,Y)Z = Z Z Z,
X Y Y X [X,Y]
∇ ∇ −∇ ∇ −∇
and
∂ ∂ ∂ ∂
R = R( , ) , .
ijkl h ∂xi ∂xj ∂xk ∂xli
We use the same notion for complex curvature operator and denote
∂ ∂ ∂ ∂
Ri¯jk¯l =hR(∂zi,∂z¯j)∂zk,∂z¯li.
ThebasicdefinitionsandpropertiesforcurvatureofK¨ahlermetricscanbefound
in [11].
Denote Cn with complex coordinates (z ,...,z ), where z = x +√ 1y .
1 n j j j
−
DefinethestandardEuclideanmetricg ,K¨ahlerformω ,andcomplexstructure
0 0
J on Cn by
0
g = n dz 2 = n (dx2+dy2),
0 j=1| j| j=1 j j
P P
ω0 = √2−1 nj=1dzj ∧dz¯j = nj=1dxj ∧dyj, and
J = n P√ 1dz ∂ P√ 1dz¯ ∂ = n dx ∂ dy ∂ ,
0 j=1 − j ⊗ ∂zj − − j ⊗ ∂z¯j j=1 j ⊗ ∂yj − j ⊗ ∂xj
P (cid:0) (cid:1) P (cid:0) (cid:1)
noting that dz =dx +√ 1dy and ∂ = 1 ∂ √ 1 ∂ .
j j − j ∂zj 2 ∂xj − − ∂yj
Darboux’sTheoremsaysthatwecanfindlo(cid:0)calcoordinates(cid:1)nearanypointon
asymplecticmanifoldsuchthatthesymplecticstructureislikeω inCninthese
0
coordinates,whichwillbecalledDarbouxcoordinates. Becauseweneedagood
controlonthe metric aswell,we willredothe argumentto findbetter Darboux
coordinates. We first start with holomorphic normal coordinates at points in a
K¨ahlermanifold, andproceedasin[6,Prop.3.2]to converttheminto Darboux
coordinates. To meet our need, we will not only derive the leading coefficients
of the metric in this new Darboux coordinates, but also the coefficients of the
next order. More precisely, we have
Proposition 2.3. Let (M,ω,g) be a compact n-dimensional K¨ahler manifold
with associate K¨ahler form ω and let U be the U(n) frame bundle of M. Then
for small ǫ>0 we can choose a family of embeddings Υ :B M depending
p,υ ǫ
smoothly on (p,υ) U, where B is the ball of radius ǫ about 0 i→n Cn, such that
ǫ
∈
for all (p,υ) U we have:
∈
(i) Υ (0)=p and dΥ =υ :Cn T M;
p,υ p,υ 0 p
| →
(ii) Υ Υ γ for all γ U(n);
p,υ γ p,υ
◦ ≡ ◦ ∈
(iii) Υ∗p,υ(ω)=ω0 = √2−1 nj=1dzj ∧dz¯j; and
P
(iv) Υ∗p,υ(g)=g0+21 Re Ri¯jk¯l(p)zizkdz¯jdz¯l +51 Re Ri¯jk¯l,m(p)zizkzmdz¯jdz¯l +
52 Re Ri¯jk¯l,m¯(Pp)zizk(cid:0)z¯mdz¯jdz¯l +O |z|(cid:1)4 , wPhere g(cid:0)0 = nj=1|dzj|2. (cid:1)
P (cid:0) (cid:1) (cid:0) (cid:1) P
6
Proof. Given(p,υ) U,wecanfindholomorphiccoordinatesthatisanembed-
∈
ding Υ′p,υ :Bǫ′ →M satisfying (i),(ii), and
(Υ′p,υ)∗(g)=g0− Ri¯jk¯l(p)zkz¯ldzidz¯j +O(|z|4)
X
i,j,k,l
1
− 2 Ri¯jk¯l,m(p)zkz¯lzm+Ri¯jk¯l,m¯(p)zkz¯lz¯m dzidz¯j.
i,jX,k,l,m(cid:0) (cid:1)
The pull back K¨ahler form has similar corresponding expression, and Υ is
′p,υ
smooth in p,υ. As in the proofof [6, Prop.3.2], we can use Moser’s method [8]
forprovingDarboux’TheoremtomodifythemapsΥ toΥ withΥ (ω)=
′p,υ p,υ ∗p,υ
ω0. Define closed 2-forms ωps,υ on Bǫ′ for (p,υ) ∈ U and s ∈ [0,1] by ωps,υ =
(1−s)ω0+s(Υ′p,υ)∗(ω). Thenthereexistafamilyof1-formsζp,υ onBǫ′ satisfying
√2−1dζp,υ =ω0−(Υ′p,υ)∗(ω), which can be taken as
1 1
ζp,υ =4Ri¯jk¯l(p)zkz¯l −z¯jdzi+zidz¯j + 10Ri¯jk¯l,m(p)zkz¯lzm −z¯jdzi+zidz¯j
(cid:0) (cid:1) (cid:0) (cid:1)
1
+ 10Ri¯jk¯l,m¯(p)zkz¯lz¯m −z¯jdzi+zidz¯j +O(|z|5).
(cid:0) (cid:1)
We use the convention that repeated indices stand for a summation whenever
there is no confusion. In the following we compute the first term of dζ as an
p,υ
example to demonstrate the argument,
1
d 4Ri¯jk¯l(p)zkz¯l −z¯jdzi+zidz¯j
(cid:0) (cid:0) (cid:1)(cid:1)
1 1 1
=− 4Ri¯jk¯l(p)z¯lz¯jdzk∧dzi− 4Ri¯jk¯l(p)zkz¯jdz¯l∧dzi− 4Ri¯jk¯l(p)zkz¯ldz¯j ∧dzi
1 1 1
+ 4Ri¯jk¯l(p)z¯lzidzk∧dz¯j + 4Ri¯jk¯l(p)zkzidz¯l∧dz¯j + 4Ri¯jk¯l(p)zkz¯ldzi∧dz¯j
1 1
=+ 4Ri¯lk¯j(p)zkz¯jdzi∧dz¯l+ 4Ri¯jk¯l(p)zkz¯ldzi∧dz¯j
1 1
+ 4Rk¯ji¯l(p)ziz¯ldzk∧dz¯j + 4Ri¯jk¯l(p)zkz¯ldzi∧dz¯j
=Ri¯jk¯l(p)zkz¯ldzi∧dz¯j.
In the second equality we use Ri¯jk¯l(p) = Rk¯ji¯l(p) = Ri¯lk¯j(p) which is implied
by the K¨ahler condition, and the last equality follows by changing the indices.
The other terms can be computed similarly, noting that there are two z¯ with
dzi and three z with dz¯j which make the coefficient from 1 to 1.
10 2
Nowletvps,υ betheuniquevectorfieldonBǫ′ withvps,υ·ωps,υ = √2−1ζp,υ. Ifwe
denote vs = 2Re as,j ∂ = Re(as,j) ∂ +Im(as,j) ∂ , the coefficient
p,υ j p,υ∂zj j p,υ ∂xj p,υ ∂yj
as,j will be P P
p,υ
1 1 1
4Ri¯jk¯l(p)zkz¯lzi+ 10Ri¯jk¯l,m(p)zkz¯lzmzi+ 10Ri¯jk¯l,m¯(p)zkz¯lz¯mzi+O(|z|5).
7
For0<ǫ6ǫ′ weconstructafamilyofembeddingsϕsp,υ :Bǫ →Bǫ′ withϕ0p,υ =
id:Bǫ →Bǫ ⊂Bǫ′ by solvingthe system ddsϕsp,υ =vps,υ◦ϕsp,υ. By compactness
of[0,1] U,thisispossibleprovidedǫ>0issmallenough. Then(ϕs ) (ωs )=
× p,υ ∗ p,υ
ω for all s, so that (ϕ1 ) (Υ ) (ω) =ω . The j-th componentof ϕ1 in z
0 p,υ ∗ ′p,υ ∗ 0 p,υ
coordinates is (cid:0) (cid:1)
1 1 1
zj+ 4Ri¯jk¯l(p)zkz¯lzi+ 10Ri¯jk¯l,m(p)zkz¯lzmzi+ 10Ri¯jk¯l,m¯(p)zkz¯lz¯mzi+O(|z|5).
(9)
Define Υ =Υ ϕ1 . ThenΥ depends smoothly onp,υ. Directcompu-
p,υ ′p,υ◦ p,υ p,υ
tations give
1
Υ∗p,υ(g)=g0+ 2 Re Ri¯jk¯l(p)zizkdz¯jdz¯l
i,Xj,k,l (cid:0) (cid:1)
1
+ 5 Re Ri¯jk¯l,m(p)zizkzmdz¯jdz¯l
i,jX,k,l,m (cid:0) (cid:1)
2
+ 5 Re Ri¯jk¯l,m¯(p)zizkz¯mdz¯jdz¯l +O |z|4 . (10)
i,jX,k,l,m (cid:0) (cid:1) (cid:0) (cid:1)
The different coefficients 1 and 2 in (10) comes from the fact that their corre-
5 5
sponding terms in (9) respectively have one z¯and two z¯. The rest of the proof
is the same as [6, Prop. 3.2], and we refer to the proof there for details.
Remark 2.4. TheK¨ahlermanifoldM doesnotneedtobecompactifweallow
ǫ depending on points.
3 Approximate examples with estimates
For 0 < t 6 R 1ǫ, consider the dilation map t : B B mapping t :
− R ǫ
→
(z ,...,z ) (tz ,...,tz ). Then Υ t is an embedding B M, so
1 n 1 n p,υ R
7→ ◦ →
we can consider the pullbacks (Υ t) (ω) and (Υ t) (g). Define a Rie-
p,υ ∗ p,υ ∗
◦ ◦
mannianmetricgt onB bygt =t 2(Υ t) (g). Itdepends smoothlyon
p,υ R p,υ − p,υ◦ ∗
t (0,R 1ǫ] and (p,υ) U, and satisfies
−
∈ ∈
t2
gpt,υ =g0+ 2 Re Ri¯jk¯l(p)zizkdz¯jdz¯l
i,Xj,k,l (cid:0) (cid:1)
t3
+ 5 Re Ri¯jk¯l,m(p)zizkzmdz¯jdz¯l
i,jX,k,l,m (cid:0) (cid:1)
2t3
+ 5 Re Ri¯jk¯l,m¯(p)zizkz¯mdz¯jdz¯l +O t4|z|4 . (11)
i,jX,k,l,m (cid:0) (cid:1) (cid:0) (cid:1)
Since t 2(Υ t) (g) is compatible with t 2(Υ t) (ω), we have that gt
− p,υ ◦ ∗ − p,υ ◦ ∗ p,υ
is compatible with the fixed symplectic form ω on B for all t,p,υ. Moreover,
0 R
8
there are uniform estimates on these metrics, which are summarized in the
following proposition.
Proposition 3.1. There exist positive constants C ,C ,C ,... such that for
0 1 2
all t (0,1R 1ǫ] and (p,υ) U, the metric gt = t 2(Υ t) (g) on B
∈ 2 − ∈ p,υ − p,υ ◦ ∗ R
satisfies the estimates
gt g 6C t2 and ∂kgt 6C tk+1 for k =1,2,..., (12)
k p,υ− 0kC0 0 k p,υkC0 k
where norms are taken w.r.t. g , and ∂ is the Levi-Civita connection of g .
0 0
Proof. This is the same as [6, Prop. 3.4]. But since we have a better estimate
on the metric from Proposition 2.3, we can increase the order on t by 1.
We can assume n a2 = 1 for simplicity. The image (Υ t)(Tn )
j=1 j p,υ ◦ a1,...,an
isaLagrangiancontPainedina B ballatpinM. Sincethe geometryofB (p)
2t 2t
in (M,ω,g) is the same as (B ,ω ,gt ) in Cn, we will do all the computations
2 0 p,υ
and discussions in (B ,ω ,gt ) instead for simplicity. In the coordinates zj =
2 0 p,υ
rje√−1θj, j =1,...,n, the metric gpt,υ becomes
gt = (dr2+r2dθ2)+ (t2ReA +t3ReC )(dr dr r r dθ dθ )
p,υ i i i ij ij i j − i j i j
X X
+ (t2ImA +t3ImC )(r dθ dr +r dr dθ )+O t4 z 4 , (13)
ij ij i i j j i j
| |
X (cid:0) (cid:1)
where
1
Aij =Aji =2 Rp¯iq¯j(p)rprqe√−1(θp+θq−θi−θj),
X
p,q
1
Cij =Cji =5 Rp¯iq¯j,m(p)rprqrme√−1(θp+θq−θi−θj+θm)
X
p,q,m
2
+ 5 Rp¯iq¯j,m¯(p)rprqrme√−1(θp+θq−θi−θj−θm). (14)
X
p,q,m
The restriction of gt on Tn is
p,υ a1,...,an
ht = a2dθ2 a a (t2ReA +t3ReC )dθ dθ +O(t4), (15)
p,υ i i − i j ij ij i j
X X
where z has been assumed to be 1 on Tn . For simplicity, we omit the
| | a1,...,an
restriction of A and C on Tn in (15), and will denote gt by g and
ij ij a1,...,an p,υ t
ht by h when there is no confusion. A direct computation yields
p,υ t
1 t2 t3
hij = δ + ReA + ReC +O(t4),
t a2 ij a a ij a a ij
i i j i j
grirj =δ t2ReA t3ReC +O t4 z 4 ,
t ij − ij − ij | |
1 t2 t3 (cid:0) (cid:1)
gθiθj = δ + ReA + ReC +O t4 z 4 ,
t r2 ij r r ij r r ij | |
i i j i j (cid:0) (cid:1)
t2 t3
griθj = ImA ImC +O t4 z 4 . (16)
t −r ij − r ij | |
j j (cid:0) (cid:1)
9
Now we are ready to compute the associate d α of the initial Lagrangian
∗ t
Tn (B ,ω ,gt ), and estimate how far it is from being Hamiltonian
a1,...,an ⊂ 2 0 p,υ
stationary.
Lemma 3.2. Denote the mean curvature vector on Tn with respect to g
a1,...,an t
by H and let α =H ω = αkdθ . Then
t t t· 0 t k
P
1 ∂2Im(t2A +t3C ) 1 ∂2Re(t2A +t3C )
ij ij jj jj
d α = +
∗ t
− a a ∂θ ∂θ 2a ∂θ ∂r
Xi,j (cid:0) i j i j i i i
1 ∂Re(t2A +t3C )
+ jj jj +O(t4), (17)
2a2 ∂θ
i i (cid:1)
where A and C are as defined in (14)
ij ij
Proof. Because ω = r dr dθ and Tn is Lagrangian,it follows that
0 k k∧ k a1,...,an
αk =a hij(Γ¯ )rkP, where (Γ¯ )c is the Christoffel symbolfor the metric g .
t kX t t θiθj t ab t
i,j
A direct computation gives
a ∂Im(t2A +t3C )
αk = 1 Re(t2A +t3C )+ k Re(t2A +t3C )+ ik ik
t − − kk kk a ik ik ∂θ
Xi i(cid:0) i (cid:1)
a ∂r2Re(t2A +t3C )
+ k i ii ii +O(t4).
2 a2∂r
Xi i k
Further computation shows that the t2 and t3 terms of αk are
t
a ∂Im(t2A +t3C ) a ∂Re(t2A +t3C )
k ik ik k ii ii
B = +
k
a ∂θ 2 ∂r
X i i X k
i i
a
+ k Re(t2A +t3C ). (18)
ik ik
a
X i
i
Recall that
∂hij ∂αj 1 ∂
d∗αt =− ∂θt αjt − hitj ∂θt − 2 hitjαjt∂θ lndet((ht)kl) .
X i X i X i(cid:0) (cid:1)
Therefore,
1 ∂Re(t2A +t3C ) 1 ∂B
ij ij i
d α =
∗ t a a ∂θ − a2 ∂θ
Xi,j i j i Xi i i
1 ∂Re(t2A +t3C )
jj jj +O(t4). (19)
− 2a2 ∂θ
Xi,j i i
From (18), we have
1 ∂B 1 ∂2Im(t2A +t3C ) ∂Re(t2A +t3C )
i ij ij ij ij
= +
a2 ∂θ a a ∂θ ∂θ ∂θ
Xi i i Xi,j i j(cid:0) i j i (cid:1)
1 ∂2Re(t2A +t3C )
jj jj
+ (20)
2a ∂θ ∂r
i i i
10