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CONSTRUCTING BALANCED METRICS ON SOME FAMILIES

OF NON-KAHLER CALABI-YAU THREEFOLDS

JIXIANG FU, JUN LI, AND SHING-TUNG YAU

Abstract. We construct balanced metrics on the family of non-Kahler Calabi-Yau threefolds that are obtained by smoothing after contracting (−1,−1)-rationalcurves on Kahler Calabi-Yau threefold. As an application, we construct balancedmetrics on complex manifolds diffeomorphic to connected sum of k ≥ 2 copies ofS3 × S3.

1. Introduction

The purpose of this paper is to construct balanced metrics on a class of non-KahlerCalabi-Yau threefolds.

A smooth Calabi-Yau threefold is a three dimensional complex manifold withfinite fundamental group and trivial canonical line bundle. In case it is Kahler, thesolution of the Calabi-Conjecture by the last author [25] provides a unique Ricci-flatKahler metric in each Kahler class of the threefold. Such metrics, known as Calabi-Yau metrics, are the bedrocks of geometric studies of Calabi-Yau threefolds. Incase the Calabi-Yau threefold is non-Kahler, one still likes to find some “canonical”metrics on it. One proposal is the balanced metrics. A balanced metric on a complexn-dimensional manifold is a hermitian metric whose hermitian form ω satisfyingd(ωn−1) = 0 (see [19]).

Balanced metrics are more flexible. For instance, the existence of balanced metricsis preserved under birational transformations [1]; in case the manifolds satisfy the∂∂-lemma, it is also preserved under deformation of complex structures [24]. In thispaper, we will consider the existence problem when the manifolds undergo a specialclass of transformations—smoothing after contraction.

Let Y be a smooth Kahler Calabi-Yau threefold that contains a collection ofmutually disjoint (−1,−1)-curves E1, · · · , El ⊂ Y ; these are smooth, isomorphic toP1 and have normal bundles isomorphic to the direct sum of two copies of degree−1 line bundles over them. By contracting all Ei, we obtain a singular Calabi-Yauthreefold X0. The Calabi-Yau threefold X0 can be smoothed to a family of Calabi-Yau threefolds Xt, possibly non-Kahler, when the curves Ei satisfy the criterion ofFriedman [6, 7]. The connected sum #kS

3×S3 of k copies of S3×S3 for any k ≥ 2can be given a complex structure in this way [7, 17]. The main theorem of thispaper is that we can find balanced metrics on Xt so that they form a well-bahavedfamily.

Theorem 1. Let Y be a smooth Kahler Calabi-Yau threefold and let Y → X0 be acontraction of mutually disjoint (−1,−1)-curves. Suppose X0 can be smoothed to a

1

family of smooth Calabi-Yau threefolds Xt. Then for sufficiently small t, Xt admitsmooth balanced metrics.

This theorem will play an important role in investigating the geometry of Calabi-Yau threefolds within the framework of Reid’s conjecture. To shed lights on the im-mense collection of diverse Calabi-Yau threefolds, Reid conjectured that all Calabi-Yau threefolds fit into a single universal moduli space in which families of smoothCalabi-Yau’s are connected by certain birational transformations and smoothings[22]. Were this confirmed, it would provide us a mean to study the geometry ofCalabi-Yau uniformly.

The current work is our attempt to strengthen Reid’s conjecture in the frameworkof metric geometry. As a first step toward this direction, we shall look into a specialclass of contraction-smoothing transformations: the conifold transitions that arecontraction of (−1,−1)-curves followed by smoothing. In this paper, we prove thatsuch transformations can be carried out metrically within the framework of balancedmetrics.

Our construction of the family of balanced metrics allows one to investigate themetric geometry of the conifold transition further along the line of Strominger’scoupled system of supersymmetry with torsion. In short, Strominger [23] proposedthe system of a pair (ω, h) of a hermitian metric ω on a Calabi-Yau threefold Y anda hermitian metric h on a vector bundle V :

d∗ω =√−1(∂ − ∂) ln ‖Ω‖ω;

F (h) ∧ ω2 = 0; F (h)2,0 = F (h)0,2 = 0;√−1∂∂ ω = 4−1α′ (tr

(

R(ω) ∧R(ω))

− tr(

F (h) ∧ F (h)))

1.

It was observed that the first equation is equivalent to that ω is conformal to abalanced metric [15]:

d(‖ Ω ‖ω ω2) = 0.

Also, in case V is the tangent bundle TY and ω is Kahler, the system is solved bytaking the Calabi-Yau metric on Y and on TY .

This system should be viewed as a generalization of Calabi conjecture for the caseof non-Kahler Calabi-Yau manifolds.

The existence of smooth solutions of the Strominger’s system has been studiedby the authors. Using perturbation method, the second and the third author [15]constructed irreducible smooth solutions to a class of Kahler Calabi-Yaus on someU(4) and U(5) principle bundles. Shortly after, the first and the third authors[9] constructed solutions to this system on a class of non-Kahler three dimensionalmanifold, i.e., on T 2-bundles over K3 surface. Recently, Fu, Tseng and Yau [8]presented explicit solutions on T 2-bundles over the Eguchi-Hanson space.

A natural question along the line of Reid’s conjecture is to see how solutions toStrominger’s system on Y are related to that of X0 and of Xt. A positive answer tothis will play an important role in the study of Calabi-Yau geometry and to super-string theories. To achieve this, the existence of families of balanced metrics on Xt

1Here Ω is the holomorphic three-form, R(ω) is the full curvature of ω and F (h) is the hermitiancurvature of h.

2

with good limiting behavior near the singularities is essential. The results proved inthis paper provide a solution to this question.

We note that for explicit existence result, Goldstein and Prokushkin [10] con-structed balanced metrics on torus bundles over K3 surfaces and over complexabelian surfaces (cf. [5] and [3]). Later, D. Grantcharov, G. Grantcharov and Y.S.Poon [11] constructed CYT structures on torus bundles over more general compactKahler surfaces; as a consequence they constructed CYT structures on complexmanifolds of topological type (k− 1)(S2 × S4)#k(S3 × S3) for k ≥ 1. Note that forcompact complex manifolds with trivial canonical line bundles, the existence of aCYT strctures and of balanced metrics are equivalent (e.g. [16]). In comparison, ourconstruction provides balanced metrics on a larger class of threefolds; they includethose of types #kS

3 × S3 for any k ≥ 2.

Corollary 2. #kS3 × S3 for any k ≥ 2 admits a balanced metric.

We now outline the proof of our existence theorem. Our first step is to modify aKahler metric on Y near the (−1,−1)-curves Ei to get a balance metric ω0 on thecontraction X0 that is smooth and balanced away from the singularities of X0; nearits singularities, ω0 coincides with the Ricci-flat metric of Candelas-de la Ossa’s (see[4]). Note that this local construction only yields a balanced metric because theareas of Ei under the new metric ω0 are zero, which is impossible if ω0 is closed.This is done in Section 2.

After, we shall deform ω0 to a family of smooth balanced metrics on Xt. SinceCandelas-de la Ossa’s metrics on the cone singularity can be deformed to smoothRicci-flat metrics on the smoothing of the cone singularity, we can deform ω0 tosmooth hermitian metrics ωt that are Kahler near the singular points of X0 and arealmost balanced on Xt for small t. To get balanced metrics, we first perturb ω2

t by

Ωt = ω2t + θt + θt, dΩt = 0,

with θt = i∂µt for µt a (1, 2)-form on Xt that solves the system

∂t∂tµt = ∂tω2t and µt ⊥ωt ker ∂t∂t.

We then solve Ωt = (ωt)2. For this possible, we need to keep Ωt positive, namely we

need that the C0-norm ‖θt‖ωt approaches zero as t approaches zero.To this end, we choose γt to be solutions to the Kodaira-Spencer equation Et(γt) =

∂ω2t subject to γt ⊥ωt kerEt. It is direct to check that the solutions γt automatically

satisfy ∂tγt = 0 and µt = −i∂∗t ∂∗t γt. Applying the elliptic estimates, the L2-estimatesand the vanishing of L2-cohomology groups, we prove that limt→0 t

κ ‖θt ‖2C0= 0 for

κ > −43 . The Section 3 and 4 are devoted to developing these estimates.

We comment that the construction of the family of hermitian metrics ωt and theestimate on the perturbation terms θt gives a precise control on the local behaviorof the metrics ωt near the singularities of X0. Such information will be useful in thefurther study on how balanced metrics transform under conifold transitions.

Acknowledgement. The authors would like to thank P.-F. Gaun, J.-X. Hong,Q.-C. Ji, L. Saper, V. Tossati and Y.-L. Xin for useful discussions. The first author

3

is supported partially by NSFC grants 10771037 and 10831008, the second authoris supported partially by NSF grant DMS-0601002, and third author is supportedpartially by NSF grants DMS-0306600 and PHY-0714648.

2. Balanced metrics with conifold singularity

Let Y be a Kahler Calabi-Yau threefold and ω its Kahler metric. Let ∪Ei ⊂ Ybe a collection of disjoint union of (−1,−1) curves. By contracting Ei we obtain avariety X0 with ordinary double point singularity. In this section, by modifying the4-form ω2 we shall construct a closed, smooth positive 4-form ω2

0 on Y − ∪Ei thatdefines a balanced hermitian metric and descends to the Candelas and De la Ossacone Ricci-flat metric near the singular points of X0.

We begin with setting up the convention for the geometry of Y near the (−1,−1)-curves. First, since the construction in this section is local, we only need to considerthe case of contracting one (−1,−1)-curve. Accordingly, we let E ⊂ Y be the(−1,−1)-curve to be contracted and let p ∈ X0 be the only singular point. We letL be the degree −1 line bundle on E; we pick a neighborhood U of E in Y that isbiholomorphic to the disk bundle in L⊕2.

To give a coordinate to U , we fix an isomorphism E ∼= P1, pick an ∞ ∈ E and letz ∈ E −∞ = C be the standard coordinate of C. We also let (u, v) be the obviouscoordinate for L⊕2|E−∞ ≡ C

⊕2E−∞, and let r be the function

(2.1) r(z, u, v) = (1+ | z |2) 12 (| u |2 + | v |2) 1

2 .

A direct check shows that this function extends to a smooth hermitian metric of L⊕2.Since the neighborhood U is biholomorphic to a disk bundle of L⊕2, we might aswell make it the unit disk bundle under the given hermitian metric. In the following,we shall fix such an isomorphism once and for all; for 1 ≥ c > 0, we shall view

(2.2) U(c) = (z, u, v) ∈ U |r(z, u, v) < c ⊂ U(1) = U ⊂ Y

as an open subset of U ⊂ Y without further mentioning.Next, we recall the Candelas-de la Ossa’s metric on U . We comment that using

(2.1) and the convention (2.2), r is a function on U ⊂ Y . We consider

i∂∂r2 =i(| u |2 + | v |2) dz ∧ dz + i(1+ | z |2)(du ∧ du+ dv ∧ dv)+ izu du ∧ dz + izu dz ∧ du+ izv dv ∧ dz + ivz dz ∧ dv.

To make the forthcoming manipulation more tractable, we notice that since bothL⊕2 and r2, thus the above form as well, are invariant under Aut(E) = PGL(2,C),to study the positivity of invariant form we only need to work out its restriction toa single point in E, say at z = 0.

We also introduce

λ1 = dz, λ2 =udu+ vdv

√

| u |2 + | v |2, λ3 =

vdu− udv√

| u |2 + | v |2and λkl = iλk ∧ λl.

Then the above form restricting to 0 is

(2.3) i∂∂r2|z=0 = r2λ11 + λ22 + λ33.

4

For the same reason, i∂r2 ∧ ∂r2 is also invariant; its restriction at z = 0 is

(2.4) i∂r2 ∧ ∂r2|z=0 = r2λ22.

Definition 3. Let f0 = 32(r

2)23 , an invariant function on U . Then the two-form

i∂∂f0 is the Kahler form of the Candelas-de la Ossa’s metric on U \E.

We denote this metric by ωco,0, abbreviated to CO-metric. In explicit form,

(2.5) ωco,0|z=0 = (r2)23λ11 + 2/3 (r2)−

13λ22 + (r2)−

13λ33.

Our next step is to modify ω by gluing the CO-metric onto ω. For this, we needto select a cut off function χ(s).

Lemma 4. There is a constant C1 such that for sufficiently large n, we can find asmooth function χ : [0,∞) → R such that

(1) χ(s) = s when s ∈ [0, 243 ];

(2) χ′(s) ≥ −C1n− 11

3 and 2χ′(s) + sχ′′(s) ≥ −C1n− 11

3 when s ∈ [243 , (n− 1)

43 ];

(3) χ′(s) ≥ −C1n− 5

3 and 2χ′(s) + sχ′′(s) ≥ −C1n− 5

3 when s ∈ [(n− 1)43 , n

43 ];

(4) χ is constant when s ≥ n43 .

Proof. We first construct a C2-function χ that satisfies the required properties. We

let c1 = 243 ; we define χ(s) = s for s ∈ [0, c1] and consider the cubic polynomial

φ(s) = c1 + (s− c1)− (s− c1)3.

Our choice of φ makes χ and φ having identical derivatives up to second order at c1.We then let c2 to be the smallest element in [c1,∞) so that 2φ′(c2) + c2φ

′′(c2) = 0.This way,

φ′(s) > 0 and 2φ′(s) + sφ′′(s) ≥ 0, for s ∈ [c1, c2].

We define χ(s) = φ(s) for s ∈ [c1, c2].

Next, we pick c3 = (n − 1)43 ; notice that for n large, c3 > c2. We define χ over

[c2, c3] to beχ(s) = χ(c2) + c2χ

′(c2)− c22χ′(c2) · s−1.

Thereforeχ′(s) > 0 and 2χ′(s) + sχ′′(s) = 0, for s ∈ [c2, c3].

To extend χ to [c3, c4] with c4 = n43 , we will do the following. We let

ψ(s) = a0 + a1(s− c3) + a2(s− c3)2 + a3(s− c3)

3;

we choose ai so that ψ(c3) = χ′(c3), ψ′(c3) = χ′′(c3) and ψ(c4) = ψ′(c4) = 0. Solvingexplicitly and using τ = c22χ

′2(c2), we get

a0 = τc−23 , a1 = −2τc−3

3 , a2 =τ(4c4 − 7c3)

c33(c4 − c3)2, a3 =

2τ(2c3 − c4)

c33(c4 − c3)3.

Using the explicit form of c3 and c4, we see that there is a constant C1 independent

of n so that for large n, −C1n− 10

3 ≤ a2 < 0 and 0 < a3 ≤ C1n− 11

3 . Therefore, over[c3, c4] we have

ψ(s) ≥ −C1n− 5

3 and 2ψ(s) + sψ′(s) ≥ −C1n− 5

3 .

5

With the function ψ at hand, we define

χ(s) =

∫ s

c3

ψ(τ)dτ + χ(c3), s ∈ [c3, c4],

and define χ be a constant function over [c4,∞).In the end, after a small perturbation of the function χ, we obtain a smooth

function that satisfies the requirements stated. This proves the Lemma.

We now set s = n43 (r2)

23 and continue to denote f0 =

32(r

2)23 , both are functions

of (z, u, v). Using the function χ, we construct a d-closed real (2, 2)-form on Y :

Φ =3

2i∂∂

(

n−43χ(s)(i∂∂f0)

)

;

expanding, it is of the form

Φ = χ′(s)(i∂∂f0) ∧ (i∂∂f0) + 2/3 n43 (r2)−

23χ′′(s)(i∂r2 ∧ ∂r2) ∧ (i∂∂f0).

Restricting Φ to z = 0 in E, from (2.4) and (2.5), we get

n23Φ|z=0 =2/3

(

2χ′(s) + sχ′′(s))

s12λ11 ∧ λ22 + 2χ′(s)s

12λ11 ∧ λ33

+ 2/3(

2χ′(s) + sχ′′(s))

s12 r−2λ22 ∧ λ33.

Applying Lemma 4, this real (2, 2)-form Φ has the following properties:(1) over U( 2n) \E, Φ = ω2

co,0 is positive;

(2) over U(1) \ U( 2n), there is a constant C2 such that for sufficiently large n,

n23Φ|z=0 ≥ −C2n

−1∑

k 6=j

λkk ∧ λjj;(2.6)

(3) over Y \ U(1), Φ ≡ 0.In conclusion, the real (2, 2)-form Φ has compact support contained in U and

thus can be considered as a global form on Y \ E.

Next we shall investigate the restriction of ω to U . We let ι : E → Y be theinclusion and consider the restriction (pull back) ω|E = ι∗ω; it is a Kahler metricon E. With π :U → E the tautological projection induced by the bundle structureof L⊕2, the form

ωE = π∗(ω|E)is a closed semi-positive (1, 1)-form on U .

Lemma 5. There is a smooth function h of U such that

ω|U = ωE + i∂∂h.

Proof. Since [ω] = [ωE ] ∈ H2dR(U,R), there exists a real 1-form α such that ω−ωE =

dα. Since α is real, we can write α = β + β for β a (0, 1)-form. Therefore from

ω − ωE = ∂β + (∂β + ∂β) + ∂β,

6

we obtain ∂β = 0. Finally, because H0,1∂

(U,C) = 02, we can find a function g on U

such that β = ∂g. Therefore for h = −i(g − g), ω − ωE = i∂∂h.

Since i∂∂h|E = ι∗(ω− ωE) = 0, the restriction h|E = const.. Thus by subtractinga constant from h we can assume that h|E = 0. Next, since the fiber Uξ = π−1(ξ)over each ξ ∈ E of the projection π : U → E is a disk in a linear space C2 ≡ R

4, therestriction of h|Uξ

has a canonical Taylor series expansion in u, u, v and v. (Here weassume ξ ∈ E −∞.) Since h|E ≡ 0, the constant terms of these series are zero. Onthe other hand, since U is the disk bundle of L⊕2, the linear parts of these Taylorseries form a globally defined function on U that is R-linear along the fibers of π.We denote this (fiberwise R-linear) function by h1 and write h2 = h− h1.

Using h1, we now introduce another (2, 2)-form. We pick a decreasing functionσ(s) that takes value 1 when 0 ≤ s ≤ 1 and vanishes when s ≥ 4. We set t = n2r2,a function of (z, u, v), and define a real (2, 2)-form on Y :

Ψ = ω2 − i∂∂(

σ(t) · h2 ·(

2ωE + i∂∂(2h1 + h2)))

− i∂∂(

σ(t) · h1 · i∂∂h1)

.

This form satisfies

dΨ = 0, Ψ|Y \U( 2n) = ω2 and Ψ|U( 1

n) = 0.

Here the first and second follows from the definitions of Ψ and σ(t); the third followsfrom i∂∂h1 ∧ ωE = 0.

We now add a multiple of the compactly supported form Φ to Ψ:

Ω0 = Ψ+ C0n23Φ, C0 > 0.

We emphasize that the form Ω0 depends on the constant C0 and the integer n. Weshall specify their choices later.

Lemma 6. The real (2, 2)-form Ω0 is d-closed, has the form Ω0 = C0n23ω2

co,0 when

restricted to U( 1n ) \ E, and has the form Ω0 = ω2 over Y \ U .Further, for sufficiently large C0 we can find an n(C0) such that for n ≥ n(C0),

the form Ω0 is positive.

Proof. Because Φ and Ψ are both d-closed, Ω0 is d-closed too. By the definitions ofΦ and Ψ, we know that

Ω0|X\U = Ψ|X\U = ω2 and Ω0|U( 1n) = C0n

23Φ|U( 1

n) = C0n

23ω2

co,0

both are positive. So we only need to check the positivity of Ω0 over U \U( 1n). We

first look at the region U( 2n) \ U( 1n). Within this region,

(2.7)Ψ =(1− σ(t))ω2 − i

(

h1∂∂σ(t) + ∂σ(t) ∧ ∂h1 + ∂h1 ∧ ∂σ(t))

∧ i∂∂h1− i

(

h2∂∂σ(t) + ∂σ(t) ∧ ∂h2 + ∂h2 ∧ ∂σ(t))

∧(

2ωE + i∂∂(2h1 + h2))

.

Since 1− σ(t) ≥ 0, the first term is non-negative. For the other two terms, becauseE is covered by D = |z| ≤ 2 and D′ = |z| ≥ 1, we only need to investigate the

2By Dolbeault theorem, this is equivalent to the vanishing of the Cech cohomology H1(U,O) = 0.The vanishing of the later follows from R1π∗O = 0 and H1(E, π∗O) = 0, where the later followsfrom that E has normal bundle (−1,−1) in U .

7

positivity over D and D′ separately. Because the discussion is similar, we shall dealwith D now.

To begin with, for a small δ we consider Vδ = π−1(D)∩U(δ). Over Vδ, the secondterm in (2.7) is

−i(

h1∂∂σ(t) + ∂σ(t) ∧ ∂h1 + ∂h1 ∧ ∂σ(t))

∧ i∂∂h1,which, after expanding, becomes

−n2h1(

σ′i∂∂r2+tσ′′r−2i∂r2∧ ∂r2)

∧i∂∂h1−tσ′r−2(i∂r2∧ ∂h1+i∂h1∧ ∂r2)∧i∂∂h1.Over the same region, we expand the relevant terms:

∂r2 = Γ−2r2zλ1 + Γrλ2

for Γ , (1+ | z |2) 12 , and

i∂∂r2 = Γ−2r2λ11 + Γ2(λ22 + λ33) + Γ−1r(zλ12 + zλ21).

For simplicity, we introduce more notations. For h1, we write a = hu(z, 0, 0) andb = hv(z, 0, 0), both are smooth functions of z; we then write the fiberwise linearfunction h1 = au + au + bv + bv. For the partial derivatives of a and b, we shall

adopt the convention az = ∂a∂z and bzz = ∂2b

∂z∂z ; namely, we use subindex z and z todenote the partial derivatives with respect to z and z. We further introduce

c11 = 2Re(azzu+ bzzv

r

)

, c21 = c12 = Γ · azu+ bzv

r, c31 = c13 = Γ · az v − bzu

r,

d12 =azu+ bzv + azu+ bz v

r, d22 = Γ · au+ bv

r, d32 = Γ · av − bu

r,

where Re is the real part. Following such convention, we have

∂h1 = rd12λ1 + d22λ2 + d32λ3

and

i∂∂h1 = rc11λ11 + c21λ21 + c12λ12 + c31λ31 + c13λ13.

To simply further, we introduce

α12 =α21 = −nh1σ′Γ2c21 + t12σ′Γc31d32;

α22 =− nh1t12σ′Γ2c11 + 2tσ′Γ−2Re(zc13d32);

α23 =α32 = nh1t12 (σ′ + tσ′′)Γ−1zc31 + tσ′Γ−2

(

zc31d22 + zc12d32)

+ tσ′Γ(c31d12 − c11d32);

α13 =α31 = −nh1(σ′ + tσ′′)Γ2c31 − t12σ′Γ(2c31Red22 − c21d32);

α33 =− nh1t12 (σ′ + tσ′′)

(

Γ2c11 − 2Γ−1Re(zc12))

− 2tσ′Γ(

c11Red22 − Re(c21d12)− Γ−3Re(zc12d22))

.

Because for r small, |u|, |v| ≤ 2r, we can find a constant depend on δ so that

|cij |, |dij | ≤ C3 for (z, u, v) ∈ Vδ.

8

To control the terms αij , we need to bound the term nh1. For this, since n−1 ≤r < 2n−1 over U( 2n) \ U( 1n), the term nh1 is bounded from above uniformly over

Vδ ∩(

U( 2n) \U( 1n))

for n > 2δ . Therefore, enlarge C3 if necessary and for n > 2

δ , wehave

|αij | ≤ C3, (z, u, v) ∈ V[ 1n, 2n] , π−1(D) ∩

(

U(2/n) \ U(1/n))

.

Finally, we introduce Λij

Λij = (iλk ∧ λk) ∧ (iλl1 ∧ λl2) = λkk ∧ λl1 l2 for i, k, l1 = j, k, l2 = 1, 2, 3.Simplifying it using the notations introduced, the expression

− i(

h1∂∂σ(t)∂σ(t) ∧ ∂h1 + ∂h1 ∧ ∂σ(t))

∧ i∂∂h1= n

∑

l=2,3

(α1lΛ1l + αl1Λl1) +∑

k,l=2,3

αklΛkl.(2.8)

We now look at the third term in (2.7). This time we consider

− i(h2∂∂σ(t) + ∂σ(t) ∧ ∂h2 + ∂h2 ∧ ∂σ(t))|Vδ

=− r−2h2(tσ′i∂∂r2 − t2σ′′r−2i∂r2 ∧ ∂r2)− tσ′r−2(∂r2 ∧ ∂h2 + ∂h2 ∧ ∂r2).

Since restricting to E the partial derivatives of h2 with respect to u and v are zero,when r is small, |h2| = O(r2) and |∂h2| = O(r). Also notice that the mixed termsuch as λ23 can be controlled by λ22 and λ33. Therefore for n > 2

δ , over V[ 1n, 2n] we

have

C3

3∑

k=1

λkk ≥ −i(h2∂∂σ(t) + ∂σ(t) ∧ ∂h2 + ∂h2 ∧ ∂σ(t)) ≥ −C3

3∑

k=1

λkk.

Therefore the third term in (2.7) can be controlled by −C3∑

k Λkk. Inserting thisand (2.8) into (2.7), we get

Ψ ≥ n∑

l=2,3

(α1lΛ1l + αl1Λl1) +∑

k,l=2,3

αklΛkl − C3

3∑

k=1

Λkk.

On the other hand by a directly calculation, we have

n23Φ|V

[ 1n , 2n ]=4/3 t−

23Γ4n2Λ11 + 4/3 t−

16nΓzΛ21 + 4/3 t−

16nΓzΛ12

+2t13 (1− 3−1Γ−2|z|2)Λ22 + 4/3 t

13 (1− Γ−2|z|2)Λ33.

Combining above two, over V[ 1n, 2n] we finally obtain

Ω0 ≥(4Γ4n2

3t23

C0 −C3

)

Λ11 + n(4Γz

3t16

C0 + α12

)

Λ12 + nα13Λ13

+n(4Γz

3t16

C0 + α21

)

Λ21 +(

2t13(

1− |z|23Γ2

)

C0 − C3 + α22

)

Λ22 + α23Λ23

+ nα31Λ31 + α32Λ32 +(4

3t

13(

1− |z|2Γ2

)C0 − C3 + α33

)

Λ33.

9

We now prove that we can find a sufficiently large constant C0 so that for anyn > 2

δ , the right hand side of the above inequality is positive. We let eij be thecoefficient of the term Λij in the above inequality. To prove the mentioned positivity,we only need to check that under the stated constraint, the three minors of the 3×3matrix [eij ] are positive:

e11 > 0, det[eij ]1≤i,j≤2 > 0, det[eij ]3×3 > 0.

We recall that t = n2r2 and Γ = (1 + |z|2) 12 . So in the region V[ 1

n, 2n], 1 ≤ t < 2

and 1 ≤ Γ ≤√5. Therefore by expanding the determinants, we see immediately

that they are all positive for n positive and C0 large enough. We fix such a C0 inthe definition of Ω0. Therefore, for any n >

2δ , the form Ω0 is positive in the region

U( 2n) \ U( 1n).

It remains to consider the region U \U( 2n). Over this region, we shall prove thatΩ0 is positive when n is large enough. For this purpose, we will use the smoothhomogenous Candelas-de la Ossa’s metric [4] on U :

(2.9) ωco = i∂∂f(r2) + i∂∂ log(1 + |z|2),

where f is defined via f ′ = r−2η for η2(η + 3/2) = r4. Explicitly,

(2.10) ωco|z=0 = (η + 1)λ11 +2

3

(η + 32 )

12

η + 1λ22 +

1

(η + 32)

12

λ33.

By simple estimate,

ω2co|z=0 ≥

1

3

∑

k 6=j

λkk ∧ λjj

Comparing with (2.6), since both ω2co and Φ are homogeneous, over U \U( 2n ) we get

n23Φ ≥ −3C2n

−1ω2co.

Therefore, over U \ U( 2n),

Ω0 ≥ ω2 − 3C0C2n−1ω2

co.

This proves that for the fixed C0 and C2, we can choose n big enough so that thereal (2, 2)-form Ω0 is positive over U \ U( 2n). This proves the lemma.

The closed (2, 2)-form Ω0 is positive (2, 2)-form on Y \ E. From [19], there is apositive (1, 1)-form ω0 on Y \ E such that ω2

0 = Ω0. This proves

Proposition 7. Let Y be a Calabi-Yau manifold and ω its Kahler metric. LetE ⊂ Y be a (−1,−1)-curve in Y . For the open subset E ⊂ U ⊂ Y chosen and forsufficiently large C0 and n, we can find a balanced metric ω0 over Y \ E such that

ω0 = ω over Y \U(1); that ω0 = C120 n

13ωco,0 over U( 1n) \E, and that ω2

0 is ∂∂-exact

over U(1) \ U( 1n).

10

Corollary 8. Let Y be a Calabi-Yau manifold and ω its Kahler metric. Let E ⊂ Ybe a union of mutually disjoint (−1,−1)-curves Ei ⊂ Y . For each Ei ⊂ Y we choosean open neighborhood Ei ⊂ Ui ⊂ Y as given by Proposition 7, and we let U be theunion of Ui. Then the Proposition 7 holds true.

Proof. Since the proof of Lemma 6 is by modifying ω2 within the open neighborhoodE ⊂ U ⊂ Y , if we choose Ei ⊂ Ui ⊂ Y to be mutually disjoint, then we can modifyω2 over the union U of Ui to obtain the desired metric ω0. Note that from the proofof Lemma 6, we can choose a common n and C0 for all i.

Because Y − E = X0,sm, the smooth part of X0, the metric ω0 descends to asmooth balance metric on X0,sm that is equivalent to the Candelas-de la Ossa’smetric near the singular points of X0.

3. constructing balanced metrics on the smoothings

Assuming the threefold X0 can be smoothed to a family of smooth Calabi-Yauthreefolds Xt, in this section we shall show that we can deform the metric ω0 to afamily of smooth balanced metrics ωt on Xt.

Definition 9. We say Xt is a smoothing of X0 if there is a smooth four dimensionalcomplex manifold X and a holomorphic projection X → ∆ to the unit disk ∆ in C

so that the general fibers Xt = X ×∆ t are smooth and the central fiber X ×∆ 0 isthe X0 we begin with.

From now on, we assume that Xt is a smoothing of X0 as defined with X the totalspace of the smoothing. We let ω0 be the balanced metric on X0,sm constructed inthe previous section.

We begin with the local geometry of X near a singular point of X0. Let p ∈ X0

be any singular point that is the contraction of E = π−1(p). Since X0 is a contrac-tion of (−1,−1)-curves in Y , from the classification of singularities of threefolds, aneighborhood of p in the total family X → ∆ is isomorphic to a neighborhood of 0in

w21 + w2

2 + w23 + w2

4 − t = 0 (in C4 ×∆),

as a family over t ∈ ∆. More precisely, for some ǫ > 0 and for

U = (w, t) ∈ C4 ×∆ǫ : ‖w‖< 1, w2

1 + w22 + w2

3 + w24 − t = 03,

there is a holomorphic map

Φ : U −→ Xcommuting with the projections U → ∆ǫ and X → ∆ so that U = Φ(U) is an open

neighborhood of p ∈ X and Φ induces an isomorphism from U to U ⊂ X .We fix such an isomorphism Φ; we denote by Ut the fiber of U over t ∈ ∆ǫ, and

denote by Ut = Φ(Ut) that is the open subset Xt ∩ U . For any 1 > c > 0, we let

U(c) = (w, t) ∈ U |‖w‖< c and Ut(c) = Φ(U(c)) ∩Xt.

3 As usual, ‖w‖2=P4

i=1 |wi|2.

11

This way, for fixed t, Ut(c) forms an increasing sequence of open subsets of Xt andthe variables (w1, · · · , w4) can be viewed as coordinate functions with the constraint∑

w2i = t understood.

In case t = 0, we can choose Φ so that the (w1, · · · , w4) relates to the coordinate(z, u, v) of (2.2) by

w1 = R(v − zu), w2 = −iR(v + zu), w3 = −iR(u− zv), w4 = R(u+ zv)

for a constant R to be determined momentarily. Hence under Φ the function r

introduced in Section 2 coincides with the function R−1r(w, 0) with r(w, 0) =‖w‖.We then define r on U to be r = rΦ−1; they are extensions of the similarly denotedr on X0 used in the previous section. Also, the punctured opens U0(c)

∗ = U0(c)− pare isomorphic to the opens U(c)−E used in the previous section under Φ as well.Since we need to work with different fibers Xt simultaneously, we shall reserve thesubscript Ut(c) to denote open subsets in Xt.

We now choose R. By choosing R large and rescaling ω0, we can assume that for

f0 =32(r

2)23

(3.1) Ω0|U0(1) = ω20|U0(1) = i∂∂(f0 · i∂∂f0).

Here since f0 is understood as a function onX0, the partials ∂ and ∂ are holomorphicand anti-holomorphic differentials of f0 over X0.

One more convention we need to introduce before we move on. Note that X0

has several singular points, say p1, · · · , pl, corresponding to contracting Ei ⊂ Y .For each such pi, we will go through the same procedure as we did for a generalsingular p ∈ X0 moments earlier to pick an open pi ∈ U ⊂ X , an isomorphismΦ : U → Φ(U) ⊂ X and the open subsets Ut(c) ⊂ Xt, etc. In fixing these Φ forvarious pi, since we can choose a common C0 and n for all i ∈ 1, · · · , l in corollary8, we can pick a single large enough R that works for all pi ∈ U so that (3.1) holdsover U .

We then form V ⊂ X (resp. V (c) ⊂ X ) that is the union of all these open subsetsU ⊂ X (resp. U(c) ⊂ X ), one for each pi ∈ X0. Accordingly, we let Vt = V ∩Xt,let Vt(c) = V (c) ∩ Xt, and let r be the function on V whose restriction to eachpi ∈ U ⊂ V is the r = r Φ−1 defined moment earlier. The particular property weshall use is that

(3.2) Ω0|V0(1) = ω20|V0(1) = i∂∂(f0 · i∂∂f0).

With these preparations, we now study the deformationXt away from the singularpoints pi ∈ X0. For c ∈ (0, 1], we introduce

Xt[c] = Xt \ Vt(c).For small t, Xt[

12 ] are diffeomorphic to each other. We fix diffeomorphisms xt :

Xt[12 ] → X0[

12 ] that depend smoothly on t and x0 = id. The diffeomorphisms xt

pulls back the form on X0[12 ] to forms on Xt[

12 ].

We then let (s) be a (decreasing) cut-off function such that (s) = 1 when s ≤ 58

and (s) = 0 when s ≥ 78 . This function define a cut off function 0 on X0 by rule

12

0|X0[1] = 0, 0|V0(12) = 1 and 0|V0(1)\V0(

12) = (r). Then

Ω0 − i∂∂(

0 · f0 · i∂∂f0)

is a smooth (2, 2)-form on X0 with compact support lies inside X0[12 ]. In particular,

for small t,x∗t

(

Ω0 − i∂∂(0 · f0 · i∂∂f0))

is a form on Xt[12 ] with compact support lies in it. So we can view this form as the

form defined on Xt by defining 0 in Vt(12).

In order to construct a positive (2, 2)-form on Xt, we need to extend the function

f0(r2) = 3

2r43 defined in Definition 3. For t 6= 0 ∈ ǫ, we define ft(s) be the function

(3.3) ft(s) = 2−13 |t| 23

∫ cosh−1( s|t|

)

0(sinh 2τ − 2τ)

13dτ, s ∈ (0, 1).

The functions ft give the Candelas-de la Ossa’s metrics (CO-metric)

Definition 10. The two form ωco,t = i∂∂ft(r2) is the Ricci-flat Kahler form on

Vt(1) constructed by Candelas and de la Ossa.

Here we clarify our convention on ∂ and ∂ over Xt. In the following, we shall takeholomorphic or anti-holomorphic differentials of functions on Xt for either t 6= 0 ort = 0. To keep the notation simple, we shall use the same ∂f and ∂f to mean thedifferentials of f on either X0 or Xt, depending on whether f is a function on X0

or Xt. We shall specifically comment on this in case there are causes for confusion.

For later application, we need to confirm the smooth dependence of the metrics

ωoc,t on t. We denote by f(k)t (s) the k-th derivative in s of ft(s).

Lemma 11. Let f0(s) =32s

23 . Then

(1). for any δ > 0 and k, restricting to s ∈ [δ, 1] the functions f(k)t (s) converges

uniformly to f(k)0 (s) when t goes to zero;

(2). For any pair 0 < δ′ < δ < 14 , there exists a αδ′ such that when | t |< αδ′ and

s ∈ [δ′, δ], 12 ≤ f ′

t(s)f ′0(s)

≤ 2 and 12 ≤ f ′′

t (s)f ′′0 (s)

≤ 2.

Proof. Since the dependence on t ∈ ∆ǫ is via its norm, we shall substitute |t| by thepositive real variable u and define fu(s) as in (3.3) with t replaced by u > 0.

At first we consider the convergence of fu(s). By L’Hospital rule, we computefrom (3.3):

limu→0

fu(s) =3

2s

23 limu→0

gu(s),

where gu(s) was defined as

gu(s) =

((

1− u2

s2

)12 − u2

s2cosh−1( su)

)13

(

1− u2

s2

)12

.

Since u2

s2cosh−1( su) ∼ u2

s2| ln u| when s ∈ [δ, 1), gu(s) converges uniformly to 1 in

[δ, 1) and so fu(s) converges uniformly to f0(s) =32s

23 .

13

Next we consider the first derivative. By (3.3), we compute

f ′u(s) = s−13 gu(s).

So f ′u(s) converges uniformly to f ′0(s) = s−13 in [δ, 1).

As to the second derivative, by directly computation, we have

f ′′u (s) =(

−s−1f ′u(s) +2

3s−2(f ′u(s))

−2)(

1− u2

s2)−1

.

It converges uniformly to f ′′0 (s) = −13s

− 43 .

Since for any k > 0, the k-th derivative of (1− u2

s2 )−1 in s is converges uniformly

to the zero function over s ∈ [δ, 1), applying induction proves the remainder casesof (1).

The second part of the Lemma follows form the explicit expressions of f ′u(s) andf ′′u (s). This proves the Lemma.

Our next step is to deform Ω0 to nearby fibers so that near the singular point itdeforms as the CO-metrics ωco,t. To this end, we define

t =

x∗t0 when on Xt[12 ],

1 when in Vt(12 );

and define

(3.4) Φt = x∗t(

Ω0 − i∂∂(0 · f0(r2) · i∂∂f0(r2)))

+ i∂∂(

t · ft(r2) · i∂∂ft(r2))

.

It is well-defined and is a d-closed 4-form on Xt. Since Xt is a complex manifold,Φt decomposes

Φt = Φ3,1t +Φ2,2

t +Φ1,3t .

We claim that for t sufficiently small, Φ2,2t is positive definite. Indeed, over Vt(

12),

the first term in (3.4) is trivial, thus

Φ2,2t |Vt(

12) = Φt|Vt(

12) = ω2

co,t,

which is positive. Over Xt[12 ], we argue that

(3.5) limt→0

Φt|Xt[12] = Ω0|X0[

12]

uniformly. From the expression of Φt it is clear that Φt only involves fu(s) and itsderivatives up to second order. Hence by (1) of the previous Lemma, we see thatover Vt(1) \ Vt(12 ), ft(r) and its partial derivatives up to second order all converges

uniformly to that of f0(r). Hence since X0[12 ] is compact and is disjoint from the

singular points, we have that the limit holds uniformly. In the end, since the partΦ3,1t and Φ1,3

t are trivial over Vt(12) and that the complex structure of Xt varies

smoothly in t, the part Φ1,3t and Φ3,1

t converges to zero uniformly as t → 0. Thisproves that limit (3.5) converges uniformly. Consequently, for sufficiently small ǫ,

Φ2,2t is positive on Xt[

12 ] for |t| < ǫ. Combined with the positivity of Φ2,2

t over Vt(12),

we obtained the desired positivity of Φ2,2t for t sufficiently small.

We let ωt be the hermitian form on Xt such that (ωt)2 = Φ2,2

t . Note that forsmall t, these metrics have uniform geometry on Xt[

12 ] and are Kahler over Vt(

12).

14

In the following we will use ωt as our background metric on Xt. Therefore objectssuch as norms and volume forms on Xt are all taken with respect to ωt.

Recall that our goal is to find balanced metrics on Xt. We shall achieve this bymodifying the form Φ2,2

t to make it both closed and positive definite.Since Φt is d-closed on Xt,

∂Φ2,2t = −∂Φ1,3

t .

We claim that for sufficiently small t, H1,3(Xt,C) = 0. Indeed, by Dolbeault theormand Serre duality, H1,3(Xt,C) = H3(Xt, T

∗Xt

) = H0(Xt, TXt). Thus to prove the

vanishing of H1,3 we suffice to prove that H0(TXt) = 0 for sufficiently small t.We now prove the vanishing of vector fields of Xt for sufficiently small t. Let ∆

be a small disk in C and let π : X∆ → ∆ be the total family of threefolds Xt fort ∈ ∆. We form the relative tangent sheaf TX∆/∆, which is defined by the the exactsequence of sheaves of OX∆

-modules:

0 −→ TX∆/∆ −→ TX∆−→ π∗T∆,

where the third arrow is induced by the projection X∆ → ∆. Since X∆ is smooth,the middle term is locally free and the third arrow is surjective away from thesingular points of X0. Therefore, TX∆/∆ is flat over ∆ and TX0 = TX∆/∆|X0 istorsion free. Now suppose for infinitely many t ∈ ∆ the threefold Xt has non-trivialvector fields; then the direct image sheaf π∗TX∆/∆ 6= 0. By the flatness of TX∆/∆,

H0(TX0) 6= 0. Since TX0 is torsion free, H0(TX0,sm) 6= 0. (Here X0,sm is the smoothloci of X0.) Now let E ⊂ Y be the contracted rational curves under the projectionY → X0, namely Y − E = X0,sm, then H0(TY−E) = H0(TX0,sm) 6= 0. But thensince Y is smooth and E ⊂ Y is a codimension 2 complex submanifold, by HartogsLemma, any section inH0(TY−E) extends to a section in H0(TY ). Since Y is Kahler,all vector fields of Y are trivial vector fields. This proves that H0(TX0,sm) = 0, a

contradiction that ensures that for sufficiently small t, H0(TXt) = 0.

Therefore there are (1, 2)-forms νt on Xt such that ∂νt = −Φ1,3t . We let µt be a

(1, 2)-form on Xt such that

(3.6) i∂∂µt = −∂Φ1,3t = ∂Φ2,2

t and µt ⊥ωt ker ∂∂.

We then define

(3.7) Ωt = Φ2,2t + θt + θt, θt = i∂µt.

Therefore (3.6) implies

∂Ωt = ∂Φ2,2t + ∂(i∂µt) + ∂(−i∂µt) = 0,

and since Ωt is real, Ωt is d-closed.

The main technical result of this section is

Proposition 12. For sufficiently small t, Ωt is positive.

Once this is proved, then the hermitian form ωt defined via (ωt)2 = Ωt is a

balanced metric on Xt.

15

To prove the Proposition, we first show

Lemma 13. Suppose limt→0 ‖θt ‖C0= 0, then Ωt is positive for small t.

Proof. We let ∗t be the hodge operator associated to the hermitian metric ωt. Then

∗tΩt = ωt + ∗t(θt + θt)

and Ωt is positive if ωt + ∗t(θt + θt) is positive.Now let qt be any closed point of Xt and let (zi) be a local chart of Xt at qt so

that

ωt(qt) =√−1δijdzi ∧ dzj and ∗t (θt + θt)(qt) = ϑ =

√−1ϑijdzi ∧ dzj .

Thus ωt + ∗t(θt + θt) is positive at qt if and only if the matrix(

δij + ϑij)

1≤i,j≤3is

positive. Since ωt(qt) =√−1δijdzi ∧ dzj ,

∑

k,l

|ϑkl|2 = | ∗t (θt + θt)(qt)|2 = |(θt + θt)(qt)|2 ≤ 4|θ(qt)|2.

Thus if |θ(qt)|2 is small, the matrix(

δij + ϑij)

is positive. In particular, if the

C0-norm ‖θt ‖C0 is small, the form ∗tΩt, and hence the form Ωt, is positive.

So we only need to prove that limt→0 ‖θt ‖2C0= 0. In the next proposition, we will

prove that limt→0 tκ ‖θt ‖2C0= 0 for any κ > −4

3 .

To estimate θt, we use the 4th-order differential operator Et (first introduced in[14]) on Λ2,3(Xt):

Et = ∂∂∂∗∂∗ + ∂∗∂∂∗∂ + ∂∗∂.

Here the adjoint is defined using the hermitian metric ωt on Xt. In [14], Kodairaand Spencer proved that Et are self-adjoint, strongly elliptic of order 4, and a formφ ∈ Ω2,3(Xt) satisfying Etφ = 0 if and only if

(3.8) ∂φ = 0 and ∂∗∂∗φ = 0.

We now let γt be a solution of

(3.9) Et(γt) = −∂Φ1,3t .

We first check that −∂Φ1,3t ⊥ kerEt. Let φ ∈ kerEt, from (3.8) we have ∂∗∂∗φ = 0;

from (3.6) we have

(−∂Φ1,3t , φ) = (i∂∂µt, φ) = (iµt, ∂

∗∂∗φ)

= 0.

This implies −∂Φ1,3t ⊥ kerEt. By the theory of elliptic operators, there is a unique

smooth solution γt⊥ kerEt of (3.9).We claim that the γt and the µt defined in (3.6) are related by

(3.10) iµt = ∂∗∂∗γt and ∂γt = 0.

This can be seen as follows. From (3.6) and (3.9), we get Et(γt)− i∂∂µt = 0, which,from the definition of the operator Et, is equivalent to

∂∂(∂∗∂∗γt − iµt) + ∂∗(∂∂∗ + 1)∂γt = 0.

16

By taking the L2-norm of the left hand side, we get

(3.11) ∂∂(∂∗∂∗γt − iµt) = 0 and ∂∗(∂∂∗ + 1)∂γt = 0.

On the other hand, for any φ ∈ ker ∂∂, we have

(∂∗∂∗γt, φ) = (γt, ∂∂φ) = 0.

Since µt⊥ ker ∂∂,

(3.12) (∂∗∂∗γt − iµt)⊥ ker ∂∂.

Combining (3.11) and (3.12), we obtain ∂∗∂∗γt − iµt = 0, which is the first identityin (3.10). The second in (3.10) follows from the second equality of (3.11), since

0 =

∫

Xt

〈∂∗(∂∂∗ + 1)∂γt, γt〉 =∫

Xt

(|∂∗∂γt|2 + |∂γt|2).

We summarize it as a Lemma

Lemma 14. We let γt be the unique solution to Et(γt) = −∂Φ1,3t subject to the

condition γt⊥ kerEt. Then γt satisfies ∂γt = 0 and the θt defined in (3.7) is of theform θt = ∂∂∗∂∗γt.

Because of this Lemma, we can apply elliptic estimate to bound the norm of γt bythat of ∂Φ1,3

t . We first check that for any given 0 < c < 12 , Et converges uniformly

to E0 on X0[c]. Since Et depends on the complex structure of Xt and the hodge staroperator of background metric ωt, it depends on ωt the derivatives of it componentsof order no more than four. By Lemma 11 and the discussion following the Lemma,for c > 0, over Xt[c] the hermition forms ωt converges to ω0 in C

4. Then because forany 0 < c < 1

2 and t sufficiently small, the Riemannian manifolds with boundaries(Xt[c], ωt) have uniform geometry, there is a constant C independent of t small sothat

(3.13) ‖γt ‖L24

(

Xt[2c])≤ C

(

‖γt ‖L2(

Xt[c]) + ‖∂Φ1,3

t ‖L2(

Xt[c])

)

.

To proceed, we argue that the quantity∫

Xt|∂Φ1,3

t |2 is bounded by C|t|2 for some

constant C. Indeed, using the explicit expression (3.4) and ∂Φ1,3t =

(

∂Φt

)2,3, we see

that ∂Φ1,3t only depend on the diffeomorphisms xt and on f0. Since we can choose xt

be smooth in t for |t| < ǫ small, and because the complex structures of Xt is smooth

in t away from the nodes of X0, the form ∂Φ1,3t is smooth in t. Then because Φ0 is

of type (2, 2), and because ∂Φt has compact support contained in Xt[12 ], we have

(3.14) supw∈Xt

‖∂Φ1,3t (w)‖< C|t|.

This provides a bound on the last term in the inequality (3.13).

Proposition 12 will follow from the following stronger estimate.

Proposition 15. For any κ > −43 ,

limt→0

(

|t|κ supXt

|θt|2ωt

)

= 0.

17

Proof. First, according to Sobolev imbedding theorems, since Xt[18 ] have uniform

geometry and Et converges uniformly to E0 on X0[18 ], there is a constant C inde-

pendent of t so that for p > 6,

|γt|C3(Xt[1/4]) ≤ C(

||γt||L2(Xt[1/8]) + ||∂Φ1,3t ||Lp(Xt[1/8])

)

Because of the identities in Lemma 14 and the inequality (3.14), there is a constantC independent of t so that

supXt[

14]

|θt|2 ≤ C(

|t|2 +∫

Xt[18]|γt|2

)

.

Multiplying by |t|κ on both sides, we get

(3.15) limt→0

(

|t|κ supXt[

14]

|θt|2)

≤ C limt→0

|t|κ∫

Xt

|γt|2.

This provides us the bound we need over Xt[14 ].

To control that over its complement, namely that inside Vt(14), we need the fol-

lowing two Lemmas whose proofs will be postponed until next section.

Lemma 16. There is a constant C independent of t such that

supVt(

14)

|θt|2 ≤ C

∫

Vt(14)|θt|2r−4 + C sup

Xt[14]

|θt|2.

Lemma 17. There is a constant C independent of t such that∫

Vt(14)|θt|2r−

83 ≤ C

∫

Xt[18](|γt|2 + |∂Φ1,3

t |2).

We continue the proof of Proposition 15. Until the end of this section, all constantC’s are independent of t; also when it depends on some other date, like a choice ofδ > 0, we shall use C(δ) to indicate so.

Since r2 ≥ |t| over Vt(1), Lemma 17 implies∫

Vt(14)|θt|2r−4 ≤ C1|t|−

23

∫

Xt[18](|γt|2 + |∂Φ1,3

t |2).

Combined with Lemma 16, we have

supVt(

14)

|θt|2 ≤ C2|t|−23

∫

Xt[18](|γt|2 + |∂Φ1,3

t |2) + C supXt[

14]

|θt|2.

Then multiplying |t|κ to both sides and taking limit t → 0, we find that by (3.14)the second term on the right hand vanishes since −2

3 + κ > −2, and the third onecan be controlled by the first one in view of (3.15). So we get

limt→0

(

|t|κ supVt(

14)

|θt|2)

≤ C3 limt→0

|t|− 23+κ

∫

Xt

|γt|2.

18

Therefore by (3.15) and the above inequality, should Proposition 15 fail we musthave

limt→0|t|−23+κ

∫

Xt

|γt|2 > 0.

In this case, there is a positive α > 0 and a sequence ti → 0 such that

|ti|−23+κ

∫

Xti

|γti |2 = α2i ≥ α2.

Normalizing γti = t− 1

3+κ

2i α−1

i γti , γti satisfy

(3.16) Eti(γti) = −t−13+κ

2i α−1

i ∂Φ1,3ti

and

(3.17)

∫

Xti

|γti |2 = 1.

Since −13 +

κ2 > −1, (3.14) implies that the C0-norm of the right hand side of (3.16)

uniformly goes to zero when i → ∞. Therefore by passing through a subsequence,there exists a smooth (1, 3)-form γ0 on X0,sm

4 such that E0(γ0) = 0 and γti → γ0pointwise.

To make sure that the limit is non-trivial, we check that ‖ γ0 ‖L2> 0. For this, weneed the following estimate that will be proved in the next section.

Lemma 18. For any 0 < ι < 13 , there is a constant C such that for any 0 < δ < 1

4and small t (|t| < δ),

∫

Vt(δ)|γt|2r−

43 ≤ Cδ2ι

∫

Xt[18](|γt|2 + |∂Φ1,3

t |2).

We continue our proving that ‖ γ0 ‖L2> 0. By (3.14) and (3.17), for large i

(3.18)

∫

Vti(δ)

|γti |2r−43 ≤ C4δ

2ι

∫

Xti[ 18](|γti |2 + t

− 23+κ

i α−2i |∂tiΦ1,3

ti|2) ≤ C5δ

2ι.

Letting i→ ∞ and using Lemma 11(2), we get

(3.19)

∫

V0(δ)∗|γ0|2r−

43 ≤ C5δ

2ι,

where V0(δ)∗ = V0(δ)\p1, · · · , pl. Because of (3.17) and the pointwise convergence

γti → γ0 over X0,sm, we have∫

X0,sm

|γ0|2 ≥ 1− C5δ2ι;

since δ is arbitrary, we obtain

(3.20)

∫

X0,sm

|γ0|2 = 1.

4X0,sm is the smooth loci of X0.

19

To obtain a contradiction to complete the proof of Proposition 15, we now showthat γ0 = 0. We first show that ∂∗γ0 = 0. Since ∂γt = 0,

Et(γt) = ∂∂∂∗∂∗γt;

consequently,∫

Xt

|∂∗∂∗γt|2 =∫

Xt

〈Et(γt), γt〉.

Substituting γti and applying the Holder inequality, (3.17), (3.16) and (3.14), weobtain

∫

Xti

|∂∗∂∗γti |2 ≤(

∫

Xti

|γti |2)

12(

∫

Xti

|Eti(γti)|2)

12 ≤ C6|ti|

23+κ

2 .

Taking i→ ∞ and noticing κ > −43 , we get ∂∗∂∗γ0 = 0.

We next pick a cut-off function τ(s) that vanishes when s ≤ 0 and τ(s) = 1 whens ≥ 1. For any 0 < δ < 1, we put sδ = 2r−δ

δ . (Note that r is a function on V0(1)

defined in (2.1) and is equal to r Φ−1.) We define

τδ = τ(sδ).

It vanished in a small neighborhood of the singular points of X0 in X0; it is aconstant function one near the boundary of V0(1). Therefore it can be extended toa function on X0. We still denote this extension by τδ. Using (2.4) and (2.5), overV0(δ) \ V0( δ2 ) and for a constant C7 independent of δ, we then estimate

(3.21) |∂τδ|2ωco,0=

4

δ2|τ ′(s)|2|∂r|2ωco,0

≤ C7r− 4

3 .

We now fix a δ1 <18 . Since τδ1∂

∗γ0 has compact support, we can view τδ1∂∗γ0 as

a (1, 3)-form on Y . Since H1,3(Y,C) = 0, there exists a smooth (1, 2)-form ςδ1 on Ysuch that

τδ1∂∗γ0 = ∂ςδ1 .

Then for any δ < δ12 , by integration by parts and using ∂∗∂∗γ0 = 0,

(3.22)∫

X0

τδ1 |∂∗γ0|2 =∫

X0

τδτδ1 |∂∗γ0|2 =∫

X0

τδ〈∂∗γ0, ∂ςδ1〉 =∫

X0

〈∗(∂τδ ∧ ∗∂∗γ0), ςδ1〉.

By Holder inequality and the definition of τδ, the right hand obeys

(3.23)

∫

X0

〈∗(∂τδ ∧ ∗∂∗γ0), ςδ1〉 ≤(

∫

V (δ)\V ( δ2)|∂τδ|2|∂∗γ0|2

)12(

∫

V (δ)\V ( δ2)|ςδ1 |2

)12.

We then apply the following estimate to be proved in the next section:

Lemma 19. For any 0 < ι < 13 , there is a constant C such that for any 0 < δ < 1

4and any small t (|t| < δ),

∫

Vt(δ)|∂∗γt|2r−

43 < Cδ2ι

∫

Xt[18](|γt|2 + |∂Φ1,3

t |2).

20

From this Lemma, we obtain for large i,

∫

Vti(δ)

|∂∗γti |2r−43 < C8δ

2ι

∫

Xti[ 18](|γti |2 +

|∂Φ1,3ti

|2

α2i |ti|

23−κ

) ≤ C8δ2ι,

where C8 is independent of δ. Taking limit i→ ∞ and using Lemma 11(2), we get∫

V0(δ)∗|∂∗γ0|2r−

43 ≤ C8δ

2ι.

Above inequality and (3.21) imply that

(3.24)

∫

V0(δ)\V0(δ2)|∂τδ|2|∂∗γ0|2 ≤ C9δ

2ι.

Next, we denote by U(δ) the union of all neighborhoods Ui(δ) of Ei in Y , usedin the previous section for 0 < δ < 1. Over V0(1)

∗ ∼= U(1) \ ∪li=1Ei we have three

metrics:

ωe = i∂∂r2, ωco,0 = i3

2∂∂(r2)

23 and ωco.

(Recall that ωco,0 is the cone Ricci-flat metric and ωco is the Ricci-flat metric onU(1) (see (2.9)). Via isomorphism Φ, Φ∗(ωe) = i∂∂r2 is a metric induced from theEuclidean metric on C

4.) Since all these metrics are homogeneous, to compare themwe only need to compare their restrictions over a single point in one Ei, say at z = 0.

Now comparing metric ωco,0 with ωe by (2.5) and (2.3), and comparing the metricsωe with ωco by (2.3) and (2.10), since ςδ1 is a (1, 2)-form, the second factor in (3.23)fits into the inequalities(3.25)

∫

V0(δ)\V0(δ2)|ςδ1 |2 ≤ C

∫

V0(δ)\V0(δ2)|ςδ1 |2ωe

volωe ≤ C10

∫

V0(δ)\V0(δ2)|ςδ1 |2ωco

r−4volωe .

Since ςδ1 and ωco are smooth on U(δ1), there exists a constant C11(δ1) possiblydepending on δ1 such that

maxU(δ1)

|ςδ1 |2ωco≤ C11(δ1).

Therefore∫

V0(δ)\V0(δ2)|ςδ1 |2ωco

r−4volωe ≤ C11(δ1)

∫

r=1

∫ δ

δ2

rdrdS ≤ C12(δ1)δ2,

where dS is the volume element of surface r = 1. Combined with (3.25), we get∫

V0(δ)\V0(δ2)|ςδ1 |2 ≤ C13(δ1)δ

2.

Then combined with (3.24) and (3.23), we obtain∫

X0

〈∗(∂τδ ∧ ∗∂∗γ0), ςδ1〉 ≤ C14(δ1)δ1+ι,

21

and with (3.22),∫

X0

τδ1 |∂∗γ0|2 ≤ C15(δ1)δ1+ι.

Taking δ → 0 and then δ1 → 0, we get∫

X0,sm|∂∗γ0|2 = 0; hence ∂∗γ0 = 0.

It remains to show that γ0 = 0. Since ∂γt = 0, we have ∂γ0 = 0. Let ϕ0 be thecomplex conjugate ¯γ0|V0(

14)∗ . Then ∂ϕ0 = ∂∗ϕ0 = 0. On the other hand, comparing

the metrics (2.3) and (2.5), and then using (3.19), we have∫

V0(14)∗|ϕ0|2ωe

volωe ≤ C

∫

V0(14)∗|ϕ0|2r−

43 < +∞.

Therefore, ϕ0 ∈ H3,2(2)

(

V0(14)

∗, ωe

)

is an L2-Dolbeault cohomology class of V0(14 )

∗,with respect to ωe.

We claim that this cohomology group vanishes. First, for any 0 < δ < 14 , V0(δ)

∗ =

V0(δ) \ p1, · · · , pl. If we let V0(δ) = Φ−1(V0(δ)), then V0(δ) is a disjoint union of l

copies of U0(δ),

U0(δ) =

(w1, · · · , w4) ∈ C4|

4∑

i=1

w2i = 0, r < δ

.

Let ωe = i∂∂r2 on U0(δ)∗ = U0(δ) \ 0 be the metric induced by the flat metric

on C4. From [21], we have limδ→0H

3,2(2)

(

U0(δ)∗, ωe

)

= 0. Since ωe = Φ∗(ωe) via the

isomorphism Φ and since V0(δ)∗ is a disjoint union of l connected open sets each of

which is isomorphic to U0(δ)∗, we also have limδ→0H

3,2(2)

(

V0(δ)∗, ωe

)

= 0. Therefore,

there exists a δ2 <14 and a (3, 1)-form ν0 on V0(δ2)

∗ such that ∂ν0 = ϕ0 and

(3.26)

∫

V (δ2)∗|ν0|2ωe

volωe < +∞.

Letϕδ2 = ϕ0 − ∂

(

(1− τδ2)ν0)

.

Then ϕδ2 has a compact support in X0,sm and ∂ϕδ2 = 0. We can view ϕδ2 as a(3, 2)-form on Y . Since H3,2(Y,C) = 0, there exists a smooth function νδ2 on Ysuch that ϕδ2 = ∂νδ2 .

Now for any δ < δ2, since ∂∗ϕ0 = 0,

(3.27)

∫

X0,sm

τδ|ϕ0|2 =∫

X0,sm

τδ〈ϕ0, ϕ0 − ∂((1− τδ2)ν0) + ∂((1− τδ2)ν0)〉

=

∫

X0,sm

τδ〈ϕ0, ∂(νδ2 + (1− τδ2)ν0)〉

=

∫

X0,sm

〈∗(∂τδ ∧ ∗ϕ0), νδ2 + (1− τδ2)ν0〉

≤(

∫

V0(δ)\V0(δ2)|∂τδ|2|ϕ0|2

)12(

∫

V0(δ)\V0(δ2)|νδ2 |2 + |ν0|2

)12.

22

Applying (3.19) and (3.21), and adding ϕ0 = ¯γ0|V0(14)∗ , we obtain

∫

V0(δ)\V0(δ2)|∂τδ|2|ϕ0|2 ≤ C16δ

2ι,

where C16 does not depend on δ. On the other hand, since νδ2 is the smooth formin Y and ωco is a smooth metric on U(δ2), there exists a constants C17(δ2) possiblydepending on δ2 such that maxU(δ2) |νδ2 |2ωco

≤ C17(δ2). This and (2.3), (2.5) and(2.10) imply that

∫

V0(δ)\V0(δ2)|νδ2 |2 ≤ C18

∫

V0(δ)\V0(δ2)|νδ2 |2ωco

r−103 volωe ≤ C19(δ2)δ

73 .

Applying (3.26), we also have∫

V0(δ)\V0(δ2)|ν0|2 ≤ C

∫

V0(δ)\V0(δ2)|ν0|2ωe

r23volωe ≤ C20.

Substituting above three inequalities into (3.27), we get∫

X0,sm

τδ|ϕ0|2 ≤ C21(δ2)δι.

Taking δ → 0, since we have fixed δ2, we obtain∫

X0,sm|ϕ0|2 = 0. This proves γ0 = 0,

a contradiction that proves the Proposition 15, and hence the Proposition 12.

4. Proofs of Lemmas 16 to 19

We shall prove Lemma 16, 17, 18 and 19 in this section.We first recall some of the notations introduced in the previous section. In the last

section, we have introduced subsets Ut ⊂ C4, the biholomorphic map Φ : Ut → Ut

and Vt ⊂ Xt, which is the union of l connected components each biholomorphic toUt.

We let ft(s) be as defined in (3.3) and let ωco,t , i∂∂ft(r2) be the corresponding

CO-metric on Ut. By definition, Φ∗(ωco,t) = ωco,t. Thus to study the metric ωco,t

we only need to investigate that of ωco,t.

One property of ωco,t we need is the following. For any c such that |t| 12 < c < 1,

the surface r = c ⊂ Ut is diffeomorphic to S2×S3 and q = (√r2−t√2, i

√r2−t√2, 0, t

12 ) is

a point in this surface. In the appendix, we will prove that we can find a holomorphiccoordinate (z1, z2, z3) at this point such that CO metric has the form

(4.1) ωco,t|q = i∂∂ft(r2)|q = i

3∑

j=1

dzj ∧ dzj .

Under this coordinate, we also have

(4.2) ∂∂r2|q = (r2)13

( r4

η3t (r2)

)13(

dz1 ∧ dz1 +3

2

η3t (r2)

r4dz2 ∧ dz2 + dz3 ∧ dz3

)

,

23

and

(4.3) ∂r2 ∧ ∂r2|q =3

2(r2)

43

(η3t (r2)

r4

)23(

1− t2

r4

)

dz2 ∧ dz2.

Here ηt(s) = sf ′t(s). In the appendix we will also prove that r−4η3t is increasing over[|t|,+∞) and

(4.4) limr2→|t|

r−4η3t =2

3, lim

r2→∞r−4η3t = 1.

We let Rijkl be the curvature tensor of ωco,t at q in coordinate (z1, z2, z3). Wehave the following Lemma estimate

Proposition 20. There exists a constant C independent of t and r such that thecurvature tensor Rijkl of the CO metric ωco,t at q are bounded by

|Rijkl| ≤ Cr−43 .

Proof. We shall prove this in the Appendix.

Let ωe , i∂∂r2 on Ut be the induced metric from Euclidean metric in C4. Let

the norm and volume form defined by this metric be | · |ωe and volωe . Comparing(4.1) with (4.2), since ωco,t and ωe are both homogeneous, we have the relation at

any point in Ut:

(4.5) volωco,t =2

3r−2volωe

and by (4.1), (4.2) and the estimate (4.4), for any smooth function f on Ut,

(4.6) |∇f |2ωe≤ Cr−

23 |∇f |2ωco,t

,

here as usual |∇f |2ω = gij ∂f∂zi

∂f∂zj

for ωij = gijdzi ∧ dzj .We comment that the prior discussion applies to metrics ωco,t on Ut(

12) since

our chosen background metric ωt restricted to Ut(12) is the CO-metric ωco,t under

the isomorphism Φ. By abuse of notation, we shall also view (z1, z2, z3) as a localcoordinate of the point Φ−1(q) ∈ Ut(

12 ). Finally, since the geometry of Vt is the

disjoint union of l copies of Ut, all statements about Ut concerning ωco,t apply to Vtas well.

For simplicity, in the following we shall adopt the following convention. Since wewill work primarily over Xt, we will omit the subscript t in all the functions andforms that was used to indicate the domain of definition. For instance, the formθt on Xt will be abbreviated to θ when the domain manifold Xt is clear from thecontext. Also, we shall continue to use ωt to be our default metric on Xt; thus allnorms and integrations without specification are with respect to the metric ωt andby the volume form of ωt. In case we need to use a different metric, say with ωe, wewill use | · |ωe and volωe to mean the associated norm and volume form.

We let τ(r) be a cut-off function defined on Vt(1) such that τ(r) = 1 when r ≤ 14

and τ(r) = 0 when r ≥ 12 . We then extend it to Xt by zero and denote by the same

notation τ(r).

24

4.1. Proof of Lemma 16. We fix a t with small |t|. As commented, we write θ tothe θt of Lemma 16.

We introduce a sequence βk = (32 )k. By the definition of τ and (4.5), we have

∫

Vt(14)|θ|2βkr−4 ≤ 2

3

∫

Vt(12)|θ|2βkr−6τ3volωe .(4.7)

The function |θ|2βkr−6τ3 is a non-negative C∞-function with compact support con-tained in Vt(

12). Via Φ, Vt(

12 ) is identified with a minimal submanifold of C4 under

the Euclidean metric. Thus we can apply the Michael-Simon’s Sobolev inequality[18] (independently by Allard [2])

(

∫

Mf

nn−1vol

)n−1n ≤ C(n)

∫

M(|∇f |+ |H| · f)vol,

where f is a nonnegative functions with compact support on an n-dimensional sub-manifold M ⊂ R

m, H the mean curvature vector of M . Here all metrics and normsare taken under those induced from the Euclidean metric on R

m.Applying this to the minimal surface Ut ⊂ C

4 and that Vt is a union of Ut, forany nonnegative function f on Vt(

12) with compact support, the above inequality

implies(

∫

Vt(12)f3volωe

)13 ≤ C

(

∫

Vt(12)|∇f |2ωe

volωe

)12,

where C is a constant depending only on the dimension of Vt(12 ). Using the volume

comparison (4.5) and the norm comparison (4.6) to the right hand side of the aboveinequality, we get

(4.8)(

∫

Vt(12)f3volωe

)13 ≤ C1

(

∫

Vt(12)|∇f |2ωco,t

r43volωco,t

)12,

for C1 a constant independent of t.We remark that in this section we shall use all Ci to denote constants that do not

depend on t and k. Since the exact size of these constants are irrelevant, we shallbe very lose in keeping track of them.

Applying (4.8) to the right hand side of (4.7) for f = |θ|2βk3 r−2τ , we have

(4.9)

(

∫

Vt(14)|θ|2βkr−4

)23 ≤

(

∫

Vt(12)|θ|2βkr−6τ3volωe

)23

≤C21

∫

Vt(12)

∣

∣∇(|θ|2βk3 r−2τ)

∣

∣

2r

43

≤3C21

∫

Vt(12)

∣

∣∇|θ|2βk3

∣

∣

2r−

83 τ2 + 3C2

1

∫

Vt(12)|θ|2βk−1 |∇r−2|2r 4

3 τ2+

+ 3C21

∫

Vt(12)|θ|2βk−1r−

83 |∇τ |2.

25

We can use (4.3) to estimate the second term in last line of the above inequality:∫

Vt(12)|θ|2βk−1 |∇r−2|2r 4

3 τ2 ≤ C2

∫

Vt(12)|θ|2βk−1r−4τ2

≤ C2

∫

Vt(14)|θ|2βk−1r−4 + 44C2

∫

Xt[14]|θ|2βk−1 .

(4.10)

From the definition of τ , the third term is controlled∫

Vt(12)|θ|2βk−1r−

83 |∇τ |2 ≤ C3

∫

Xt[14]|θ|2βk−1 .(4.11)

It remains to estimate the first term in the last line of (4.9). We claim that forany k ≥ 1,

(4.12)

∫

Vt(12)

∣

∣∇|θ|2βk3

∣

∣

2r−

83 τ2 ≤ −C4

∫

Vt(12)|θ|2βk−1 ∂ (r

− 83 τ2)−

− βk−1

∫

Vt(12)|θ|2(βk−1−1)gβα(〈∇α∇βθ, θ〉+ 〈θ,∇α∇βθ〉)r−

83 τ2.

We first prove the case k ≥ 3. By direct calculation, we have

(4.13)

∫

Vt(12)

∣

∣∇|θ|2βk3

∣

∣

2r−

83 τ2 =

β2k−1

4

∫

Vt(12)|θ|2(βk−1−2)

∣

∣∇|θ|2∣

∣

2r−

83 τ2.

Using the definition ∂ = −gβα ∂2

∂zα∂zβ, we compute

βk−1(βk−1 − 1)|θ|2(βk−1−2)∣

∣∇|θ|2∣

∣

2

=βk−1|θ|2(βk−1−1) ∂ |θ|2 −∂ |θ|2βk−1

≤− βk−1|θ|2(βk−1−1)gβα(〈∇α∇βθ, θ〉+ 〈θ,∇α∇βθ〉)−∂ |θ|2βk−1 .

Multiplying r−83 τ2 to both sides of above inequality and then integrating over Vt(

12),

since the CO metric is Kahler and τ2 vanishes outside Vt(12), we get

βk−1(βk−1 − 1)

∫

Vt(12)|θ|2(βk−1−2)

∣

∣∇|θ|2∣

∣

2r−

83 τ2

≤− βk−1

∫

Vt(12)|θ|2(βk−1−1)gβα(〈∇α∇βθ, θ〉+ 〈θ,∇α∇βθ〉)r−

83 τ2

−∫

Vt(12)|θ|2βk−1 ∂

(

r−83 τ2

)

.

This and (4.13) proves (4.12).

For k = 2, from ∂ |θ|3 = 32 |θ| ∂ |θ|2 − 3|θ|

∣

∣∇|θ|∣

∣

2, a computation gives

∫

Vt(12)

∣

∣∇|θ|2β23

∣

∣

2r−

83 τ2 ≤β1

∫

Vt(12)|θ|2(β1−1) ∂ |θ|2r−

83 τ2 −

∫

Vt(12)|θ|2β1 ∂ (r

− 83 τ2).

This implies (4.12) in case of k = 2.

26

For k = 1, we need to estimate∣

∣∇|θ|∣

∣

2. When |θ| = 0, | |θ|| = 0. When |θ| 6= 0,

then

∣

∣∇|θ|∣

∣

2=1

4|θ|−2

∣

∣∇|θ|2∣

∣

2 ≤ 2−1gβα〈∇αθ,∇βθ〉+ 2−1gβα〈∇βθ,∇αθ〉

=− 2−1 ∂ |θ|2 − 2−1gβα〈∇α∇βθ, θ〉 − 2−1gβα〈θ,∇α∇βθ〉.

So (4.12) is still valid for k = 1 and β0 = 1.Next we estimate the second term in (4.12) by using the Kodaira’s Bochner for-

mula ([20] p.119): for any (p, q)-form ψ =∑

ψα1···βqdzα1 ∧ · · · ∧ dzβq ,

(∂ψ)α1···βq=−

∑

α,β

gβα∇α∇βψα1···βq

+

p∑

i=1

q∑

k=1

∑

α,β

Rα β

αiβkψα1···αi−1ααi+1···βk−1ββk+1···βq

−q

∑

k=1

∑

β

R ββkψα1···βk−1ββk+1···βq

.

(4.14)

We use above formula to ψ = θ over Vt(12 ). Since∂θ|Vt(

12) = 0 and from Proposition

20 the curvature is bounded by Cr−43 , we have

− gβα(〈∇α∇βθ, θ〉+ 〈θ,∇α∇βθ〉)=− gβα(〈∇α∇βθ, θ〉+ 〈θ,∇β∇αθ〉+ 〈θ, [∇α,∇β]θ〉) ≤ C5r

− 43 |θ|2,

where [∇α,∇β ] = ∇α∇β −∇β∇α is the curvature operator.From the above inequality, we can estimate the second term in (4.12):

(4.15)

− βk−1

∫

Vt(12)|θ|2(βk−1−1)gβα(〈∇α∇βθ, θ〉+ 〈θ,∇α∇βθ〉)r−

83 τ2

≤ C5βk−1

∫

Vt(12)|θ|2βk−1r−4τ2

≤C5βk−1

∫

Vt(14)|θ|2βk−1r−4 + 44C5βk−1

∫

Xt[14]|θ|2βk−1 .

From (4.1), (4.2) and (4.3),

−∂ r− 8

3 ≤ C6r−4.

Thus the first term in (4.12) can be controlled

(4.16)

−∫

Vt(12)|θ|2βk−1 ∂ (r

− 83 τ2)

≤C6

∫

Vt(14)|θ|2βk−1r−4 + 44C6

∫

Xt[14]|θ|2βk−1 .

27

Inserting (4.15) and (4.16) into (4.12), we get

(4.17)

∫

Vt(14)

∣

∣∇|θ|2βk3

∣

∣

2r−

83 τ2 ≤ C7βk−1

(

∫

Vt(14)|θ|2βk−1r−4 +

∫

Xt[14]|θ|2βk−1

)

.

Then inserting (4.17), (4.10) and (4.11) into (4.9), at last we obtain(

∫

Vt(14)|θ|2βkr−4

)1βk ≤ (C7βk−1)

1βk−1

(

∫

Vt(14)|θ|2βk−1r−4 +

∫

Xt[14]|θ|2βk−1

)1

βk−1

So for any k ≥ 1, from above inequality, either(

∫

Vt(14)|θ|2βkr−4

)1βk ≤

(

2C7βk−1

)

1βk−1

(

∫

Vt(14)|θ|2βk−1r−4

)1

βk−1

or(

∫

Vt(14)|θ|2βkr−4

)1βk ≤

(

2C7βk−1

)1

βk−1(

vol(Xt[1/4]))

1βk−1 sup

Xt[14]

|θ|2

must hold. Since the volume of Xt[14 ] can be controlled by a constant independent

of t, these two inequalities imply

(

∫

Vt(14)|θ|2βkr−4

)1βk ≤

k−1∏

i=1

(C8βi−1)1

βi−1

(

∫

Vt(14)|θ|2r−4 + sup

Xt[14]

|θ|2)

,

for C8 independent of t and k. Taking limit k → ∞, we get the inequality stated inLemma 16.

4.2. Proof of Lemma 17. We shall continue working over Xt and the opens Vt(δ),and write θ instead of θt when Xt is understood. To streamline the notation, wewill assign the symbol Λt to

Λt :=

∫

Xt[18](|γ|2 + |∂Φ1,3|2).

The Lemma 17 is to show that for a constant C independent of t,∫

Vt(14)|θt|2r−

83 ≤ CΛt.

To begin with, for a smooth positive function φ, we define ∂∗φζ = ∂∗ζ−∗(∂ log φ∧∗ζ), ∇φ

α = ∇α + ∂α log φ and X βφ βk

= −gβkα∂β∂α log φ. We need another Kodaira’s

Bochner formula ([20], p.124): For any (p, q)-form ζ =∑

ζα1···βqdzα1 ∧ · · · ∧ dzβq ,

((∂∂∗φ + ∂∗φ∂)ζ)α1···βq=−

∑

α,β

gβα∇φα∇βζα1···βq

+

p∑

i=1

q∑

k=1

∑

α,β

Rα βαiβk

ζα1···αi−1ααi+1···βk−1ββk+1···βq

+

q∑

k=1

∑

β

(X βφ βk

−R ββk

)ζα1···βk−1ββk+1···βq.

(4.18)

28

We now let ψ = ∂∂∗γ. Since the CO metric is Kahler over Vt(12), over Vt(

12) we

have θ = ∂∂∗∂∗γ = −∂∗ψ. We apply the Kodaira formula for φ = φ1 = r−83 and

ζ = ψ. Since ψ is a (2, 3)-form and the CO metric is Ricci flat, we obtain(4.19)

∫

Vt(12)〈∂∂∗φ1

ψ,ψ〉φ1τ = −∫

Vt(12)〈gβα∇φ1

α ∇βψ,ψ〉φ1τ +∫

Vt(12)

3∑

β=1

X βφ1 β

|ψ|2φ1τ.

Since τ has a compact support in Vt(12), we compute

∫

Vt(12)〈∂∂∗φ1

ψ,ψ〉φ1τ

=

∫

Vt(12)〈∂∗φ1

ψ, ∂∗φ1ψ〉φ1τ +

∫

Vt(12)〈∂τ ∧ ∂∗φ1

ψ,ψ〉φ1

≤∫

Vt(12)|θ|2φ1τ +

∫

Vt(12)|∂ log φ1 ∧ ∗ψ|2φ1τ

− 2Re

∫

Vt(12)〈∗(∂ log φ1 ∧ ∗ψ), ∂∗ψ〉φ1τ + · · · ,

where the dots denote terms that are integrations over Xt[14 ] of smooth function

including the derivatives of τ . By (3.13), the dotted terms are controlled by a fixedmultiple, independent of t, of Λt =

∫

Xt[18](|γ|2 + |∂Φ1,3|2). In the remainder of this

section, the term CΛt will appear in various places for the same reason.On the other hand, since ∂∂∗ψ = −∂Φ1,3 = 0 on Vt(

12 ),

∫

Vt(12)|θ|2φ1τ =Re

∫

Vt(12)〈∂∗φ1

ψ, ∂∗ψ〉φ1τ +Re

∫

Vt(12)〈∗(∂ log φ1 ∧ ∗ψ), ∂∗ψ〉φ1τ

≤Re

∫

Vt(12)〈∗(∂ log φ1 ∧ ∗ψ), ∂∗ψ〉φ1τ + C1Λt.

We remark that the C1 and the Ci to appear later are all independent of t. Com-bining the above two inequalities, we get

∫

Vt(12)〈∂∂∗φ1

ψ,ψ〉φ1τ ≤ −∫

Vt(12)|θ|2φ1τ +

∫

Vt(12)|∂ log φ1 ∧ ∗ψ|2φ1τ +C2Λt.

Inserting the above inequality into (4.19) and applying divergence theorem to thefirst term on the right hand side (4.19), since ψ is a (2, 3)-form, we get

∫

Vt(12)|θ|2φ1τ ≤

∫

Vt(12)

(

|∂ log φ1|2 −3

∑

β=1

X βφ1 β

)

|ψ|2φ1τ + C3Λt.(4.20)

According to (4.1)-(4.4), we have

|∂ log φ1|2 ≤8

3r−

43 and

3∑

β=1

X βφ1 β

≥ 8

3r−

43 .

29

So from (4.20), we get

∫

Vt(14)|θ|2r− 4

3 ≤ C3Λt = C3

∫

Xt[18](|γ|2 + |∂Φ1,3|)2.

This proves Lemma 17.

4.3. Proof of Lemma 18 and Lemma 19. We first prove Lemma 18. The samemethod can be used to prove Lemma 19.

For any 0 < ι < 13 and any 0 < δ < 1

4 , by Holder inequality,

∫

Vt(δ)|γ|2r− 4

3 ≤(

∫

Vt(14)|γ|3r−3ι

)23(

∫

Vt(δ)r−4+6ι

)13.

Clearly,

(

∫

Vt(δ)r−4+6ι

)13=

(2

3

∫

Vt(δ)r−6+6ιvolωe

)13 ≤ Cδ2ι,

where constant C does not depend on t and δ. So to prove the Lemma we only needto prove that for a constant C independent of t,

(4.21)(

∫

Vt(14)|γ|3r−3ι

)23 ≤ C

∫

Xt[18](|γ|2 + |∂Φ1,3|2).

We will prove the above inequality in three steps. Our first step is to establishthe inequality

(4.22)(

∫

Vt(14)|γ|3r−3ι

)23 ≤ 8

∫

Vt(12)|∂∗γ|2r−2ιτ2 + C1

∫

Vt(12)|γ|2r−2ι− 4

3 τ2 +C1Λt.

We now prove this inequality. Using the method in deriving (4.9) and (4.12) fork = 1, we get(4.23)

(

∫

Vt(14)|γ|3r−3ι

)23 ≤ C2

∫

Vt(12)

∣

∣∇|γ|∣

∣

2r−2ιτ2+

+ C2

∫

Vt(12)|γ|2|∇r−ι− 2

3 |2r 43 τ2 +C2

∫

Vt(12)|γ|2r−2ι|∇τ |2

≤ C3

∫

Vt(12)gβα(〈∇αγ,∇βγ〉+ 〈∇βγ,∇αγ〉)r−2ιτ2

+ C3

∫

Vt(12)|γ|2r−2ι− 4

3 τ2 +C3Λt.

30

Let φ2 = r−2ι and φ3 = r−2ι− 43 . By divergence theorem,

(4.24)

∫

Vt(12)gβα

(

〈∇αγ,∇βγ〉+ 〈∇βγ,∇αγ〉)

φ2τ2

≤− 2

∫

Vt(12)gβα〈∇φ2

α ∇βγ, γ〉φ2τ2 +∫

Vt(12)〈gβα[∇β,∇α]γ, γ〉φ2τ2 + C4Λ

+

∫

Vt(12)gβα〈∂α log φ2 · ∇βγ, γ〉φ2τ2 −

∫

Vt(12)gβα〈∇αγ, ∂β(φ2τ

2)γ〉.

To bound the four terms after the identity sign, we use that the curvature is bounded

by Cr−43 to the second item, and use the Holder inequality to the last two items.

After simplification, we get

(4.25)

∫

Vt(12)gβα(〈∇αγ,∇βγ〉+ 〈∇βγ,∇αγ〉)φ2τ2

≤− 4

∫

Vt(12)gβα〈∇φ2

α ∇βγ, γ〉φ2τ2 + C4

∫

Vt(12)|γ|2φ3τ2 + C4Λt.

We now deal with the first term on the right hand side of the above inequality.We use the Kodaira’s formula (4.18) to the case ζ = γ and φ = φ2 in this subsection.Since γ is a (2, 3)-form and CO metric is Ricci flat, (4.18) reduces to

−∑

α,β

gβα∇φ2α ∇βγ = ∂∂∗φ2

γ −3

∑

β=1

X βφ2 β

γ.

So we get

−∫

Vt(12)gβα〈∇φ2

α ∇βγ, γ〉φ2τ2 =∫

Vt(12)〈∂∂∗φ2

γ, γ〉φ2τ2 −∫

Vt(12)

3∑

β=1

X βφ2 β

|γ|2φ2τ2.

By Holder inequality, we estimate∫

Vt(12)〈∂∂∗φ2

γ, γ〉φ2τ2 =∫

Vt(12)〈∂∗φ2

γ, ∂∗φ2γ〉φ2τ2 −

∫

Vt(12)〈∂τ2 ∧ ∂∗φ2

γ, γ〉φ2

≤ 2

∫

Vt(12)|∂∗γ|2φ2τ2 + 2

∫

Vt(12)|∂ log φ2|2|γ|2φ2τ2 + C5Λt.

Put together, we get

(4.26)

−∫

Vt(12)gβα〈∇φ2

α ∇βγ, γ〉φ2τ2 ≤∫

Vt(12)

(

2|∂ log φ2|2 −∑

X βφ2 β

)

|γ|2φ2τ2+

+ 2

∫

Vt(12)|∂∗γ|2φ2τ2 + C5Λt.

On the other hand, by direct calculation,

|∂ log φ2|2 ≤3

2ι2r−

43 and

∑

X βφ2 β

≥ 2ιr−43 .

31

So inequality (4.26) implies

(4.27)

−∫

Vt(12)gβα〈∇φ2

α ∇βγ, γ〉φ2τ2

≤2

∫

Vt(12)|∂∗γ|2φ2τ2 +

∫

Vt(12)(3ι2 − 2ι)|γ|2φ3τ2 + C5Λt.

Finally, inserting (4.27) into (4.25) and then inserting (4.25) into (4.23), we completeour first step in establishing the inequality (4.22).

Our second step is to prove∫

Vt(12)|γ|2φ3τ2 ≤

2

2ι− 3ι2

∫

Vt(12)|∂∗γ|2φ2τ2 + C6Λt.(4.28)

For this, we first apply the divergence theorem to the left had side of (4.27):

(2ι − 3ι2)

∫

Vt(12)|γ|2φ3τ2 ≤2

∫

Vt(12)|∂∗γ|2φ2τ2 + C6Λt.(4.29)

This inequality implies (4.28) since when ι < 13 , 2ι− 3ι2 > 0.

Our third step is to prove

(4.30)

∫

Vt(12)|∂∗γ|2φ2τ2 ≤ C7

∫

Xt[18](|γ|2+ | ∂Φ1,3 |2)).

To achieve this, we write

2

∫

Vt(12)|∂∗γ|2φ2τ2 =2Re

∫

Vt(12)〈∂∗φ2

γ, ∂∗γ〉φ2τ2

+2Re

∫

Vt(12)〈∗(∂ log φ2 ∧ ∗γ), ∂∗γ〉φ2τ2.

(4.31)

The first term after the equal sign in (4.31) is bounded by

≤2Re

∫

Vt(12)〈γ, ∂∂∗γ〉φ2τ2 + C8Λt

≤2b

∫

Vt(12)|γ|2φ3τ2 +

1

2b

∫

Vt(12)|∂∂∗γ|2φ4τ2 + C8Λt,

for some b > 0, and for φ4 = r−2ι+ 43 and φ3 = r−2ι− 4

3 . By (4.3), the second itemafter the equal sign in (4.31) is bounded by

≤∫

Vt(12)|∂∗γ|2φ2τ2 +

3

2ι2∫

Vt(12)|γ|2φ3τ2.

Therefore (4.31) implies∫

Vt(12)|∂∗γ|2φ2τ2 ≤ (

3

2ι2 + 2b)

∫

Vt(12)|γ|2φ3τ2 +

1

2b

∫

Vt(12)|∂∂∗γ|2φ4τ2 + C8Λt.

32

Inserting (4.28) into the above inequality and simplifying, we obtain

(4.32)ι(2− 6ι)− 4b

ι(2− 3ι)

∫

Vt(12)|∂∗γ|2φ2τ2 ≤

1

2b

∫

Vt(12)|∂∂∗γ|2φ4τ2 + C9Λt.

Since ι < 13 , for any given ι we can choose b such that ι(1− 3ι)− 2b > 0. Then the

above inequality implies

(4.33)

∫

Vt(12)|∂∗γ|2φ2τ2 ≤

ι(2− 3ι)

2b(

ι(1− 3ι)− 2b)

∫

Vt(12)|∂∂∗γ|2φ4τ2 +C10Λt.

Finally, we need to estimate∫

Vt(12) |∂∂∗γ|2φ4τ2. Since the COmatric is Kahler and

∂γ = ∂γ = 0, then ∂∂∗γ = ∂∂∗γ. When restricted to Vt(12 ), ∂∂

∗∂∂∗γ = −∂Φ1,3 = 0.

From these identities, since 0 < ι < 13 ,

(4.34)

∫

Vt(12)|∂∂∗γ|2r−2ι+ 4

3 τ2 ≤∫

Vt(12)|∂∂∗γ|2τ2

=

∫

Vt(12)〈∂∂∗γ, ∂∂∗γ〉τ2 ≤

∫

Vt(12)〈∂∂∗∂∂∗γ, γ〉τ2 + C11Λt = C11Λt.

Combining the above two inequalities, we prove the inequality (4.30), our third step.Inserting (4.30) into (4.28) and then inserting (4.28) and (4.30) into (4.22), we

get (4.21). Thus we finished the proof of Lemma 18.

Finally we prove Lemma 19. The proof is parallel to that of the previous Lemma,except that in Lemma 18 the form γ is a (2, 3)-form while in this Lemma ∂∗γ is a(1, 3)-form. Replacing γ by ∂∗γ, we find that all inequalities up to (4.33) are valid.

So to prove Lemma 19 we only need to estimate∫

Vt(12) |∂∂∗∂∗γ|2r−2ι+ 4

3 τ2. Since

∂∂∗∂∗γ = ∂∗∂∂∗γ, by the same method in proving (4.34), we get∫

Vt(12)|∂∂∗∂∗γ|2r−2ι+ 4

3 τ2 ≤ C12

∫

Xt[18](|γ|2 + |∂Φ1,3|2).

This proves Lemma 19.

Appendix A. Estimates on Candelas-de la Ossa’s metrics

We first recall some notations from Candelas-de la Ossa’s paper [4]. We considerthe family Vt:

Vt = (w1, · · · , w4) |4

∑

i=1

(wi)2 = t ⊂ C

4.

Since the individual Vt only depend on |t|, in the following we shall work with t > 0.

We let r2 =∑4

i=1 |wi|2 be the radial coordinate. We set

ωco,t = i∂∂ft(r2)

The condition that the metric be Ricci-flat is

(A.1) r2(r4 − t2)(η3t )′ + 3t2η3t = 2r8 with ηt(r

2) = r2f ′t(r2).

33

The scale has been chosen so ηt has the same asymptotic behavior as r43 for large

r. After settingr2 = t cosh τ, for τ ≥ 0

and integrating, we pick the solution

(A.2) ηt =2−1/3t2/3

tanh τ(sinh 2τ − 2τ)1/3.

Note that this choice of ηt makes the metric regular at r2 = t. Also note that from(A.1), f ′t(s) = s−1ηt(s), and that ft(s) defined in (3.3) is a solution of this equation.

In this appendix, we want to estimate the curvature of the CO metric. Since itis homogeneous (see [4]), we only need to perform our calculation at points q =

(√r2−t√2, i

√r2−t√2, 0, t

12 ). At first we pick some orthogonal coordinate at this point.

Since dw1 ∧ dw2 ∧ dw3 6= 0 near q, we can take (w1, w2, w3) as a (holomorphic)coordinate in a neighborhood of the point q. By directly calculation, we get

∂∂r2|q =r2 + t

2t

(

dw1∧dw1+dw2∧dw2

)

+dw3∧dw3+ir2 − t

2t

(

dw2∧dw1−dw1∧dw2

)

and∂r2 ∧ ∂r2|q = 2(r2 − t)dw2 ∧ dw2.

To simplify them, we introduce a new coordinate (u1, u2, u3) at the point q:

w1 =2t

r2q + tu1 − i

r2q − t

r2q + tu2, w2 = u2, w3 = u3,

where r2q , r2(q). Under this coordinate, the ∂∂r2 and ∂r2 ∧ ∂r2 are expressed as

(A.3) ∂∂r2|q =2t

r2 + tdu1 ∧ du1 +

2r2

r2 + tdu2 ∧ du2 + du3 ∧ du3

and∂r2 ∧ ∂r2|q = 2(r2 − t)du2 ∧ du2.

Combined with (A.1),

(A.4) f ′t + r2f ′′t = η′t =2r8 − 3t2η3t

3r2(r4 − t2)η2t;

so at the point q the CO metric is

i∂∂ft|q =2tηt

r2(r2 + t)idu1 ∧ du1 +

4r4

3η2t (r2 + t)

idu2 ∧ du2 +ηtr2idu3 ∧ du3.

At last we introduce a new coordinate (z1, z2, z3) near the point q as:

z1 =( 2tηt(q)

r2q(r2q + t)

)12u1, z2 =

( 4r4q3η2t (q)(r

2q + t)

)12u2, z3 =

(ηt(q)

r2q

)12u3.

The CO metric at this point is then expressed as

i∂∂ft(r2)|q = i

3∑

j=1

dzj ∧ dzj .

34

Under this coordinate, we can rewrite (A.3) as

∂∂r2|q = (r2)13

( r4

η3t

)13(

dz1 ∧ dz1 +3

2

η3tr4dz2 ∧ dz2 + dz3 ∧ dz3

)

and

∂r2 ∧ ∂r2|q =3

2(r2)

43

(η3tr4

)23(

1− t2

r4

)

dz2 ∧ dz2.To estimate the curvature of the CO metric, we need to investigate the asymptotic

behavior ofη3tr4.

Lemma 21. Over [t,+∞), the function r−4η3t is an increasing function and

limr2→t

r−4η3t =2

3, lim

r2→∞r−4η3t = 1.

Proof. Let h(τ) = r−4η3t . From (A.2),

h(τ) =1

2

cosh τ(sinh 2τ − 2τ)

sinh3 τ.

Differentiating,

h′(τ) =1

2 sinh4 τh1(τ) where h1(τ) = 4τ + e2τ (τ − 3/2) + e−2τ (τ + 3/2)

and

h′1(τ) = 2τe2τ − 2e2τ − 2τe−2τ − 2e−2τ + 4.

Thus for any τ > 0,

h′1(τ) = 4τ

∞∑

n=1

(2τ)2n+1

(2n+ 1)!· n

n+ 1> 0.

Since h1(0) = 0, h1(τ) > 0 and h′(τ) > 0. So over [0,+∞), the function r−4η3t is aincreasing function of τ . Since r2 = t cosh τ for τ ≥ 0 is an increasing function inτ , r−4η3 is increasing in r2. Since τ → 0 when r2 → t, and τ → ∞ when r2 → ∞,we obtain the two desired limits by applying the L’Hospital rule. This proves theLemma.

We next investigate η′t. From η3t = r4h(τ),

3η2t η′t = 2r2h(τ) + r4h′(τ)

dτ

dr2= 2r2h(τ) + t−1r4h′(τ) sinh−1 τ > 0,

hence r23 η′t > 0. On the other hand by (A.4), we get

r23 η′t =

2− 3 t2

r4η3tr4

3(1− t2

r4)(

η3tr4)23

=(η3tr4

)13 +

2− 3η3tr4

3(1 − t2

r4)(

η3tr4)23

.

Then from Lemma 21, we see that

0 < r23 η′t < 1.

35

In the following for two functions α(r, t) and β(r, t) in r and t we shall use α . βto mean that there is a constant C independent on r and t such that

|α(r, t)| ≤ C|β(r, t)|.

Under this convention, the previous Lemma and the last inequality can be abbrevi-ated as

ηt . r43 and η′t . r−

23 .

For higher derivatives, by introducing ǫ = tr2, the identities (A.1) and (A.4) imply

η′′t . r−83 (1− ǫ)−1, η

(3)t . r−

143 (1− ǫ)−2.

Therefore, by the second identity of (A.1), we obtain the following asymptotic esti-mate

f ′t . r−23 , f ′′t . r−

83 , f

(3)t . r−

143 (1− ǫ)−1 and f

(4)t . r−

203 (1− ǫ)−2.

To proceed, we need to the partial derivatives of r2 with respect to zi and zi. Forsimplicity, we shall use the subscript i to denote the partial derivative with respect

to zi, and use i for derivatives with respect to zi. Thus, for instance,∂2r2

∂zi∂zj= (r2)ij .

Under this convention, we compute directly that at the point q, the first orderpartial derivatives

(A.5) (r2)1 = (r2)3 = 0 and (r2)2 = −√6

2i(r2 − t)

12 (r2 + t)

12ηtr2

. r43 (1− ǫ)

12 ;

the second order derivatives (r2)ij = 0 except the following

(A.6) (r2)11 = (r2)33 =r2

ηt. r

23 , (r2)22 =

3

2

η2tr2

. r23 ;

the second derivatives (r2)ij = 0 except the following

(r2)11 = (r2)33 = −r2

ηt. r

23 , (r2)22 = −3

2

η2tr2ǫ . r

23 .

For the third order partial derivatives of type ijk, we have the vanishing (r2)ijk = 0except the following

(r2)111 =r3(1− ǫ)

12

t12 η

32t (1 + ǫ)

12

. ǫ−12 (1− ǫ)

12 , (r2)121 = −i

√6

2

(1− ǫ)12

(1 + ǫ)12

. (1− ǫ)12 ,

(r2)212 =3

2

t12 η

32t (1− ǫ)

12

r3(1 + ǫ)12

. ǫ12 (1− ǫ)

12 , (r2)222 = −i3

√6

4

tη3t (1− ǫ)12

r6(1 + ǫ)12

. ǫ(1− ǫ)12 ,

(r2)313 =r3(1− ǫ)

12

t12 η

32t (1 + ǫ)

12

. ǫ−12 (1− ǫ)

12 , (r2)323 = −i

√6

2

(1− ǫ)12

(1 + ǫ)12

. (1− ǫ)12 .

36

For the fourth order partial derivatives, we still have the vanishing except the fol-lowing

(r2)1111 =r4

tη2t. r

43 t−1, (r2)1212 = (r2)2121 =

3ηt2r2

. r−23 ,

(r2)2222 =9tη4t4r8

. tr−83 , (r2)1313 = (r2)3131 =

r4

tη2t. r

43 t−1,

(r2)3333 =r4

tη2t. r

43 t−1, (r2)2323 = (r2)3232 =

3ηt2r2

. r−23 .

We now use these asymptotic estimate of the partial derivatives of r2 to proveProposition 20; namely that there is a constant C independent of t and r such that

the curvature tensor Rijkl of the CO metric ωco,t at q has bound

(A.7) Rijkl . r−43 .

Since the coordinate (z1, z2, z3) is orthogonal at q, the curvature at q has the form

Rijkl = −(ft)ijkl + (ft)ikq(ft)qjl.

One group of terms appearing in (ft)ijkl are of the type

f(4)t · (r2)i(r2)j(r2)k(r2)l, f

(3)t ·

∑

(r2)i1i2(r2)i3(r

2)i4 , f ′′t ·∑

(r2)i1i2(r2)i3i4 ,

where the summations are taken over all possible permutation i1, i2, i3, i4 =i, j, k, l. For such type of terms, using the previous estimate, we check directly

that they are bounded by Cr−43 .

The other group of terms in appearing (ft)ijkl are of the type:

f ′t · (r2)ijkl(A.8)

f ′′t ·(

(r2)ijk(r2)l + (r2)ilk(r

2)j)

,(A.9)

f ′′t ·(

(r2)jkl(r2)i + (r2)jil(r

2)k)

.(A.10)

Of these, (A.8) vanishes when i 6= k or j 6= l, (A.9) vanishes when i 6= k and (A.10)vanishes when j 6= l. The remainder cases in (A.8)-(A.10) may be not vanishing,and will be treated separated momentarily.

We now look at the product term (ft)ikq(ft)qjl. First in the expression of (ft)ikq,the following two types of terms

f(3)t · (r2)i(r2)k(r2)q and f ′′t ·

∑

i1,i2,i3

(r2)i1i2(r2)i3

are bounded by Cr−23 ; therefore corresponding product terms

(

f(3)t · (r2)i(r2)k(r2)q + f ′′t ·

∑

i1,i2,i3

(r2)i1i2(r2)i3

)

×(

f(3)t · (r2)j(r2)l(r2)q + f ′′t ·

∑

j1,j2,j3

(r2)j1j2(r2)j3

)

37

in the expansion of (ft)ikq(ft)qjl are also bounded by Cr−43 . Here the summations

are over all possible permutation i1, i2, i3 = i, k, q and j1, j2, j3 = j, l, q.The remainder terms in (ft)ikq(ft)qjl are of the following types:

(f ′t)2 · (r2)ikq(r2)j lq,(A.11)

f ′t · (r2)ikq ·(

f ′′t∑

(r2)i1i2(r2)i3 + f

(3)t · (r2)j(r2)l(r2)q

)

,(A.12)

f ′t · (r2)j lq ·(

f ′′t∑

(r2)i1i2(r2)i3 + f

(3)t · (r2)i(r2)k(r2)q

)

,(A.13)

where the summation in the second line is taken over all possible permutationi1, i2, i3 = j, l, q and summation in the last line is taken over all possible per-mutation i1, i2, i3 = i, k, q. Like before, they vanish when i 6= k in case (A.11)and (A.12) or when j 6= l in case (A.11) and (A.13).

Combining the above discussion, we see that when i 6= k and j 6= l, the bound(A.7) follow immediately. For others, we need to treat case by case.

For the case i 6= k and j = l, we have

Rijkj . r−43 − f ′′t · (r2)jji(r2)k

(

1− f ′t · (r2)ii)

− f ′′t · (r2)j jk(r2)i(

1− f ′t · (r2)kk)

.

We claim that the last term is always zero. Indeed, because of (A.5), if i 6= 2, thenthe last term is equal to zero. If i = 2, then k 6= 2, and by (A.1) and (A.6) we have1 − f ′t · (r2)kk = 0. This proves the claim that the last item always vanishes. Forthe same reason, the second item vanishes. Therefore, when i 6= k and j = l, theestimate (A.7) holds.

For the case i = k and j 6= l, we have Rijil = Rjili . r−43 . This proves the bound

(A.7) in this case.Finally, we need to consider the cases i = k, j = l and i 6= j. We should consider

these cases individually. In case ijkl = 1313, we have

R1313 .r− 4

3 − f ′t · (r2)1313 + (f ′t)2 · (r2)111(r2)331

+(

f ′t · (r2)112 + f ′′t · (r2)11(r2)2)

·(

f ′t · (r2)332 + f ′′t · (r2)33(r2)2)

.

The last term is clearly . r−43 ; the second and third items combined give

−f ′t · (r2)1313+(f ′t)2 · (r2)111(r2)331 = −f ′t

r4

tη2t+(f ′t)

2 r6(r2 − t)

tη3t (r2 + t)

=−2r2

ηt(r2 + t). r−

43 .

This proves the bound (A.7) for R1313. By similar method, we obtain desired bound(A.7) for R1212 and R2323.

Finally, we consider the case i = j = k = l. Since the metric is Ricci-flat, we have

Riiii = −∑

j 6=i

Riijj . r−43 .

This completes the proof of Proposition 20.

Remark 22. We remark that when r → t, the induced metric on the surface r2 =const. approaches

1

2

(2t2

3

)13ds2|S3 ,

38

where ds2|S3 is the standard metric on S3. The curvature of the limiting metric is

Ct−23 for some constant C.

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Institute of Mathematics, Fudan University, Shanghai 200433, China

E-mail address: majxfu@fudan.edu.cn

Department of Mathematics, Stanford University, Stanford, CA94305, USA

E-mail address: jli@math.stanford.edu

Department of Mathematics, Harvard University, Cambridge, MA 02138, USA

E-mail address: yau@math.harvard.edu

40