K\"ahlerity of shrinking gradient Ricci solitons asymptotic to K\"ahler cones

0.2 MB·

by  Brett Kotschwar

KA¨HLERITY OF SHRINKING GRADIENT RICCI SOLITONS
ASYMPTOTIC TO KA¨HLER CONES
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1
BRETTKOTSCHWAR
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2

n Abstract. WeprovethatashrinkinggradientRiccisolitonwhichisasymp-
a totic to a K¨ahler cone along some end is itselfK¨ahler onsome neighborhood
J ofinfinityofthatend. Whentheshrinkeriscomplete, itisgloballyK¨ahler.

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3

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G 1. Introduction
D
Let M be a smooth manifold of dimension n. A shrinking gradient Ricci soliton
.
h structure on M consists of a Riemannian metric g and a smooth function f which
t together satisfy the equations
a
m 1
(1) Rc(g)+ f = g, R+ f 2 =f.
[ ∇∇ 2 |∇ |

1 The latter equation is a normalization which can be achieved by adding an appro-
v priate constant to f on any connected component of M.
6 ShrinkinggradientRiccisolitons(orshrinkers)correspondnaturallytoshrinking
8
self-similar solutions to the Ricci flow, which are both the generalized fixed points
4
of the equation and the prototypical models for the behavior of a solution in the
8
0 vicinity of a developing singularity. They have been fully classified in dimensions
. two [H2] and three [CCZ], [H2], [I1], [N], [NW], [P], [PW], and even a limited
1
0 extension of this classification to dimensions four and higher would help advance
7 the understanding of the long-time behavior of solutions to the equation.
1 There are indications that the asymptotic behavior of complete noncompact
:
v shrinkingsolitonsmayberigidenoughtosupportaclassificationintogeneralstruc-
i tural types. At present, all known examples of complete noncompact shrinkers
X
which are not locally reducible as products are connected at infinity and smoothly
r
asymptotic to regular cones (see [FIK], [DW], [Y]). Recent results of Munteanu-
a
Wang [MW1, MW3, MW4] suggest that these two types may actually exhaust the
possibilities for complete noncompact four-dimensional shrinkers.
In this paper, we consider shrinkers of the second type, whose geometries are
asymptotically conical in the following precise sense. Given a closed Riemannian
manifold (Σ,g ) of (real) dimension n 1 and a fixed a R, let E = E (Σ)
Σ a a
− ∈
denote the cylinder (a, ) Σ and g = dr2 +r2g the regular conical metric
c Σ
∞ ×
on E . Further, for any fixed λ > 0, let ρ : E E denote the dilation map
0 λ 0 0
→
ρ (r,σ) + (λr,σ). By an end V (M,g), we will mean an unbounded connected
λ
⊂
component of M D for some compact D M.
\ ⊂
Date:November2016.
Theauthor waspartiallysupportedbySimonsFoundationgrant#359335.
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2 BRETTKOTSCHWAR
Definition. Let V be an end of (M,g). We say that (M,g) is asymptotic to
(E ,g ) along V if, for some r > 0, there is a diffeomorphism Φ : E V such
0 c r
→
that λ−2ρ∗Φ∗g g in C2 (E ,g ) as λ .
λ → c loc 0 c →∞
Using the correspondence between shrinkers and self-similar solutions, it is pos-
sibletoshowthatanyshrinkerwhosecurvaturedecaysquadraticallyonanendwill
be asymptotic to some regular cone on that end in the above sense and that the
convergence is actually locally smooth (see, e.g., [KW] and cf. [CL]). Munteanu
and Wang [MW3] have shown that any complete shrinker whose Ricci curvature
tendstozeroatinfinitymusthavequadraticcurvaturedecayandmustthereforebe
asymptotically conical along each of its ends. In dimension four, they have proven
thatthe sameistrueassumingonlythatthe scalarcurvaturetends tozero[MW2].
The uniqueness theorem proven in [KW] asserts that if two shrinking gradient
Riccisolitonsareasymptotictothesameregularconealongsomeendofeach,then
thesolitonsarethemselvesisometriconsomeneighborhoodofinfinityoftheseends.
Oneconsequenceofthistheoremisthatanynon-trivialisometryofthecross-section
of the cone must be reflected in a non-trivial isometry of the end of the soliton.
In this paper, we show that the K¨ahlerity of the cone is another feature which is
necessarily inherited by an asymptotic shrinker.
Theorem 1. Suppose (M,g,f) is a shrinking gradient Ricci soliton asymptotic to
the regular cone (E ,g ) along the end V M. If (E ,g ) is K¨ahler with complex
0 c 0 c
⊂
structure J , then there is an end W V of M and a complex structure J defined
c
⊂
on W relative to which g is K¨ahler and (W,J) is biholomorphic to (E ,J ) for
W R c
|
some R>0. If (M,g) is complete, the K¨ahler structure extends to all of M.
If it is known separately that there is a K¨ahler shrinker (Mˆ,gˆ,fˆ) asymptotic to
(E ,g ) along some end of (Mˆ,gˆ), the first part of this theorem is a consequence
0 c
of the uniqueness result in [KW]. The point here is that the K¨ahlerity of the
soliton (M,g,f) on the end W is established only from its asymptotic K¨ahlerity
along V. No information off of V is required for this conclusion, and the soliton
need not be complete nor connected at infinity. However, when (M,g) is complete
andconnected,itsuniversalcover(M˜,g˜)willbegloballyK¨ahlerand,consequently,
connectedatinfinity[MW1]. Itfollowsthenthat(M,g)isalsoconnectedatinfinity
and that the K¨ahler structure on W must extend to all of M.
Our proof of Theorem 1 uses a combination of the methods in [K2] and [KW].
In [KW], it is shown that a shrinker which is asymptotic to a cone (E ,g ) gives
0 c
rise to a smooth solution g(t) to the Ricci flow on some end W V for t [ 1,0)
⊂ ∈ −
which begins at g( 1) = g, evolves self-similarly for 1 t < 0, and converges
− − ≤
smoothly as t 0 to a limit metric isometric to g . By this construction, recalled
c
→
in Proposition 8 below, the first assertion in Theorem 1 is reduced to a problem
of “backward propagation of K¨ahlerity,” that is, to the problem of showing that
a solution to the Ricci flow which becomes K¨ahler after some time must have
actually been K¨ahler all along. This problem was considered in [K2] within the
class of complete solutions of bounded curvature.
Whereastheproblemofthepropagationofthe K¨ahlerstructureforwardintime
can, in principle, be reduced to the uniqueness of solutions to the Ricci flow by
the constructionof K¨ahler solutions with the same initial data, the backwardtime
problemdoesnotseemtoadmitacorrespondinglystraightforwardreductiontothe
backward uniqueness of solutions. The K¨ahler solutions which one would need to
KA¨HLERITY OF SHRINKERS ASYMPTOTIC TO KA¨HLER CONES 3
construct to act as competitors are now the solutions to parabolic terminal-value
problems which are ill-posed in general. These issues are discussed in some detail
inSection2.3of[K2]inthecontextofthe preservationofholonomyalongtheflow.
The problem in [K2] is instead recast as one of backward uniqueness for a pro-
longed system encoding a certain decomposition of the curvature operator. In the
nextsection,wediscussthe specializationofthisformulationtotheK¨ahlersetting.
This sets up a problem of backward uniqueness which, analytically, is identical in
structure and setting to that consideredin [KW], and can be resolvedby the same
argument(theapplicationofappropriateCarlemaninequalities)giveninthatrefer-
ence. We discussthe applicationofthese resultstothe firstassertionofTheorem1
inSection3,andprovethesecondassertion,concerningtheextensionoftheK¨ahler
structure to the entire manifold, with a short synthetic argument in Section 4.
2. Backward propagation of Ka¨hlerity under the Ricci flow
In this section, we will assume that g is a K¨ahler metric on M with complex
0
structureJ andthatg =g(τ)isasmoothfamilyofmetricssatisfyingthebackward
0
Ricci flow
∂
(2) g =2Rc(g)
∂τ
on M [0,T] with g(0)=g . We do not assume here that g(τ) is self-similar.
0
×
Ouraimis tosetupa frameworkto determine whenthe K¨ahlerityofthe metric
g is transferred to g(τ) for τ >0. The initial value problem for (2) is ill-posed, so
0
thepropagationoftheK¨ahlerstructurecannotbesimplyreducedtotheproblemof
the uniqueness of solutions to (2) (that is, to the problem of backward uniqueness
of solutions to the Ricci flow). Instead we will recall (and specialize to the K¨ahler
setting)theapproachin[K2],whichencodesthe generalproblemofpreservationof
reduced holonomy in terms of the uniqueness of solutions to an associated system
of mixed differential inequalities.
2.1. A family of almost-complex structures on M. The first step in our re-
duction of the problem is to extend J to a family of almost complex structures
0
J =J(τ):TM TM on [0,T] relative to which g(τ) remains Hermitian. (Even-
→
tually we will wish to argue that J(τ) = J .) We do so by solving the ordinary
0
differential initial value problem
∂ Ja =RcJa RaJc on M (0,T]
(3) ∂τ b b c − c b ×
(cid:26) J(x,0)=J0(x) on M
in each fiber of End(TM).
The evolution equation in (3) may be understood in terms of the operator D
τ
which acts on families V =V(τ) of (k,l)-tensors by
∂
D Va1a2...al = Va1a2...al Rc Va1a2...al Rc Va1a2...al Rc Va1a2...al
τ b1b2...bk ∂τ b1b2...bk − b1 cb2...bk − b2 b1c...bk −···− bk b1b2...c
+Ra1Vca2...al +Ra2Va1c...al + +RalVa1a2...c.
c b1b2...bk c b1b2...bk ··· c b1b2...bk
Relative to a smooth family of local frames e (τ) n evolving so as to remain
{ i }i=1
orthonormalrelative to g(τ), the components of D V express the total derivatives
τ
∂
D Va1a2...al = V(e ,e ,...,e ,e∗ ,e∗ ,...,e∗ ) .
τ b1b2...bk ∂τ b1 b2 bk a1 a2 al
(cid:0) (cid:1)
4 BRETTKOTSCHWAR
Alternatively, D may be regardedas a vector on the product of the frame bundle
τ
with[0,T]tangenttothebundleofg(τ)-orthonormalframes. See[H1]orAppendix
F of [CRF] for details.
Equations (2) and (3) are then equivalent to the assertions that D g = 0 and
τ
D J =0, and imply that
τ
J2 = Id, g(, )=g(J ,J ),
− · · · ·
thatis, thatJ remains analmostcomplex structure andg remains Hermitianwith
respect to J on M [0,T],
×
2.2. A time-dependentsplittingof 2T∗M. Wewillnowexaminetherelation-
∧
ships between several families of endomorphisms of 2T∗M induced by J. Below,
∧
we will write
θ σ =θ σ σ θ,
∧ ⊗ − ⊗
for θ, σ T∗M, and use the metric
∈
θ,φ θ,ψ
(4) θ σ,φ ψ =det h i h i
h ∧ ∧ i (cid:18) σ,φ σ,ψ (cid:19)
h i h i
induced on 2T∗M by g(τ).
∧
Now define F, G End( 2T∗M) by
∈ ∧
1
(Fη)(X,Y)= (η(X,JY)+η(JX,Y)), (Gη)(X,Y)=η(JX,JY),
2
for η 2T∗M and X, Y in TM, and let
∈∧
1 1
P˜ = (Id+G), Pˆ = (Id G).
2 2 −
The identities
(5) G2 =Id, FG=GF= F
−
satisfied by F and G imply the following relations between F, P˜, and Pˆ.
Lemma 2. On M [0,T] we have the identities
×
P˜2 =P˜, Pˆ2 =Pˆ, PˆP˜ =P˜Pˆ =0, P˜ +Pˆ =Id,
P˜∗ =P˜, Pˆ∗ =Pˆ, F∗ = F,
−
F2 = Pˆ, P˜F=FP˜ =0, FPˆ =PˆF=F.
−
Here ∗ denotes the adjoint of the operator relative to the metric (4).
The first two rows of relations imply, among other things, that
= (τ)=imP˜(τ), = (τ)=imPˆ(τ),
H H K K
are complementary orthogonalsubbundles of 2T∗M.
∧
Lemma 3. For any (p,τ) M [0,T], we have the g(τ)-orthogonal decomposition
∈ ×
(6) 2(T∗M)= (τ) (τ)
∧ p Hp ⊕Kp
and (τ) is a subalgebra of 2(T∗M) isomorphic to u(n/2) relative to the bracket
Hp ∧ p
(7) [ω,η] =ω η ω η
ij ik kj jk ki
−
induced by g(τ).
KA¨HLERITY OF SHRINKERS ASYMPTOTIC TO KA¨HLER CONES 5
Proof. As we have noted, the first assertion follows from the first two rows of
Lemma 2. For the second assertion,let s (τ)=so(n) denote the Lie subalgebra of
p ∼
endomorphisms of T M (under commutation) which are skew-symmetric relative
p
to g(p,τ). The endomorphisms in s (τ) which further commute with J = J(p,τ)
p
compriseasubalgebrau (τ)ofs (τ)isomorphictou(n/2). ThemapΦ :s (τ)
p p p,τ p
→
2(T∗M) given by Φ (A)(X,Y) = g(X,AY) is an isomorphism relative to the
∧ p p,τ
bracket(7) on 2(T∗M), and the image of u (τ) under Φ, consisting of those two-
∧ p p
formsηforwhichη(X,Y)=η(JX,JX)forallX,Y T M,isprecisely (τ). (cid:3)
p p
∈ H
Remark 4. Relative to 2CT∗M = 2T∗M RC, we have
∧ ∧ ⊗
1 1
P(2,0) = Pˆ √ 1F , P(0,2) = Pˆ +√ 1F , P(1,1) =P˜,
2(cid:16) − − (cid:17) 2(cid:16) − (cid:17)
where P(2,0), P(0,2), P(1,1) are the projections of 2T∗M onto 2,0M, 0,2M, and
C
∧ ∧ ∧
1,1M, respectively.
∧
2.3. The decomposition of the curvature operator. Let us use R to denote
the (4,0) curvature tensor and R : 2T∗M 2T∗M to denote the curvature
∧ → ∧
operator of g(τ). We adopt the convention that
R(η) = R η , η 2T∗M.
ij ijab ab
− ∈∧
Relative to an orthonormal frame e n , we have
{ i}i=1
R =R = R(e∗ e∗),e∗ e∗ = 2R e∗ e∗,e∗ e∗ = 2R .
ijkl klij h i ∧ j k∧ li − abijh a∧ b k∧ li − ijkl
Similarly, we will use S for the operator R T∗M End( 2T∗M) such that
∇ ∈ ⊗ ∧
S(X,η) = 2 R X η
ij a ijbc a bc
− ∇
and adopt the notation
S = S(e ,e∗ e∗),e∗ e∗ = 2 R .
mijkl h m i ∧ j k∧ li − ∇m ijkl
Now define R˜, Rˆ : 2T∗M 2T∗M by
∧ →∧
R˜ =R P˜, Rˆ =R Pˆ
◦ ◦
and S˜, Sˆ T∗M End( 2T∗M) by
∈ ⊗ ∧
S˜ = R P˜, Sˆ = R Pˆ.
∇ ◦ ∇ ◦
Here,
S˜(X,η)=( R) P˜(η), Sˆ(X,η)=( R) Pˆ(η),
X X
∇ ◦ ∇ ◦
for X TM and η 2T∗M.
∈ ∈∧
Lemma 5. At τ =0, we have
(8) P˜ = Pˆ = F=0,
∇ ∇ ∇
(9) R F=0, Rˆ =0, R˜ =R,
◦
and
(10) R F=0, Sˆ =0, S˜ = R.
∇ ◦ ∇
6 BRETTKOTSCHWAR
Proof. When τ =0, we have J =0 and consequently
∇
R(X,Y,JZ,W)+R(X,Y,Z,JW)=0
for all X, Y, Z, W TM. The former implies that F= Pˆ = P˜ =0 and the
∈ ∇ ∇ ∇
latter implies that R F=0; in fact,
◦
(R F)(Z∗ W∗)(X,Y)=R(X,Y,JZ,W)+R(X,Y,Z,JW).
◦ ∧
Since Pˆ = F2 andP˜ =Id Pˆ by Lemma 2, we also haveR Pˆ =0 and R P˜ =R.
− − ◦ ◦
Then
0= (R F)= R F+R F= R F,
∇ ◦ ∇ ◦ ◦∇ ∇ ◦
and the other identities in (10) follow similarly. (cid:3)
The identities in Lemma 5 reflect that the image of the curvature operator is
contained in the parallel subalgebra hol(g(0)) of 2T∗M defined by the holonomy
∧
representationof g(0). Since g(0) is K¨ahler, this means that, at any p M,
∈
imR(p,0) hol (g(0)) (0)=u(n/2),
⊂ p ⊂Hp ∼
and so
kerR(p,0) ⊥(0)= (0),
⊂Hp Kp
since R(p,0) is self-adjoint.
2.4. A closed system of mixed differential inequalities. Now define
A= Pˆ T∗M End( 2T∗M), B= Pˆ T∗M T∗M End( 2T∗M),.
∇ ∈ ⊗ ∧ ∇∇ ∈ ⊗ ⊗ ∧
ItfollowsfromthegeneralcomputationsinSection4of[K2]thatRˆ andSˆ,together
with A and B, satisfy a closed system of differential inequalities.
Proposition 6. The sections A, B, Rˆ, Sˆ satisfy
(11) D A C R A +C Sˆ
τ
| |≤ | || | | |
(12) D B C R A +C R B + Sˆ
τ
| |≤ |∇ || | | |(cid:16)| | |∇ |(cid:17)
(13) (D +∆)Rˆ C R A +C R B + Rˆ
τ
(cid:12) (cid:12)≤ |∇ || | | |(cid:16)| | | |(cid:17)
(cid:12) (cid:12)
(14) (cid:12)(D +∆)Sˆ(cid:12) C R A +C R Rˆ +C R B + Rˆ + Sˆ
τ
(cid:12) (cid:12)≤ |∇∇ || | |∇ || | | |(cid:16)| | | | | |(cid:17)
(cid:12) (cid:12)
for some(cid:12)C =C(n) o(cid:12)n M [0,T].
×
The only properties of the projections P˜ and Pˆ of which the argument in [K2]
makes use is that they remain complementary and orthogonal, evolve according
to D P˜ = D Pˆ = 0 (which follows here from the fact that D J = 0), and that
τ τ τ
= P˜( 2T∗M) remains closed under the Lie bracket. The computations in fact
H ∧
hinge on the relations
(15) [ , ] , [ , ] ,
H H ⊂H H K ⊂K
implied by this latter condition. We will review here the derivation of (13), but
refer the reader to [K2] for the rest of the details.
Recall that, under (2), the curvature operatorevolves accordingto the equation
(16) (D +∆)R= R2 R#,
τ
− −
KA¨HLERITY OF SHRINKERS ASYMPTOTIC TO KA¨HLER CONES 7
where the second term denotes the Lie algebra square R# = R#R. Here, for M,
N End( 2T∗M), M#N End( 2T∗M) is the operator defined at p M by
∈ ∧ ∈ ∧ ∈
1
(17) (M#N)(η)= [Mϕ ,Nϕ ],η [ϕ ,ϕ ],
i j i j
2 h i
Xi,j
where ϕ n(n−1)/2 is any orthonormalbasis for 2T∗M.
{ i}i=1 ∧ p
Now, we may compute directly from (16) that
(D +∆)Rˆ = (R2+R#) Pˆ +2 R Pˆ +R ∆Pˆ,
τ i i
− ◦ ∇ ◦∇ ◦
and
(D +∆)Rˆ C R (Rˆ + B )+C R A + R# Pˆ .
τ
(cid:12) (cid:12)≤ | | | | | | |∇ || | | ◦ |
(cid:12) (cid:12)
So, to obtain(cid:12) (13), it rem(cid:12)ains only to estimate R# Pˆ.
◦
Using to denote the adjoint of an element in End( 2T∗M), we have
∗ ∧
R=Pˆ R+P˜ R=Rˆ∗+R˜∗
◦ ◦
and thus, by the symmetry and bilinearity of the # pairing, that
(18) R# Pˆ =(R#R) Pˆ =(Rˆ∗#Rˆ∗+2R˜∗#Rˆ∗) Pˆ +(R˜∗#R˜∗) Pˆ.
◦ ◦ ◦ ◦
But (R˜∗#R˜∗) Pˆ =0 since, by (15), [P˜Rϕ ,P˜Rϕ ],Pˆϕ =0 for all i, j, k. So
i j k
◦ D E
(R#R) Pˆ C(Rˆ∗ + R˜∗ )Rˆ∗ C R Rˆ
| ◦ |≤ | | | | | |≤ | || |
by (18), and (13) follows.
2.5. Preservation of K¨ahlerity from the vanishing of A and Rˆ. We have
seenthatifthe initialtime-slice (M,g(0))isK¨ahler,thenA,B, Rˆ, andSˆ vanishon
this slice. In the next section, we will show that the backwarduniqueness theorem
provenin[KW]impliesthatA,B,Rˆ,andSˆmustvanishidenticallyonM [0,T]for
×
thespecificclassofsolutions(M,g(τ))wewillencounterintheproofofTheorem1.
Fornowweobservethatthevanishingofthesesectionsindeedimplythat(M,g(τ))
will remain K¨ahler relative to the fixed complex structure J(0)=J .
0
Lemma 7. Let g, J, Rˆ, and Pˆ be as above. If Rˆ = 0 and Pˆ = 0 on M [0,T],
∇ ×
and J =0 on M 0 , then J =0 and ∂ J =0 on M [0,T].
∇ ×{ } ∇ ∂τ ×
Proof. First observe that our assumptions imply that S F=0. Indeed,
◦
S F=S (Pˆ F)=Sˆ F=( R Pˆ) F= Rˆ F=0,
◦ ◦ ◦ ◦ ∇ ◦ ◦ ∇ ◦
usingLemma2. TheobservationthatJ remainsparallelthenfollowsfromasimple
pointwise calculation. Fix p M and let Wa = Jb. Since D J =0, we have
∈ ib ∇i a τ
∂ ∂ ∂
Wa = (RaJc RcJa) Γc Ja+ ΓaJc
∂τ ib ∇i c b − b c − ∂τ ib c ∂τ ic b
=Jagcd( R R )+Jcgad( R R ) RaWc +RcWa
c ∇d bi−∇b di b ∇c di−∇d ci − c ib b ic
= gacglm R Jd+gadglm R Jc RaWc +RcWa
− ∇l mibd c ∇l midc b − c ib b ic
=gadglm( R Jc+ R Jc) RaWc +RcWa
∇l micb d ∇l midc b − c ib b ic
1
= gadglm(S F) RaWc +RcWa
2 ◦ lmibd− c ib b ic
= RaWc +RcWa.
− c ib b ic
8 BRETTKOTSCHWAR
Since W(p,0)=0, it follows that ( J)(p,τ)=W(p,τ)=0.
∇
But,ifJ isparallelandg isHermitianrelativetoJ,wethenhaveRc J =J Rc
◦ ◦
as endomorphisms of TM. So
∂
Rc=J Rc Rc J =0,
∂τ ◦ − ◦
and J(,τ)=J(,0) on M [0,T]. (cid:3)
· · ×
More generally, if g(τ) is a solution to the backward Ricci flow for which the
reduced holonomy remains fixed (e.g., if (M,g(τ)) is complete and of bounded
curvature) then any tensor V which is parallel on M with respect to g(0) can
0
be extended to a smooth family V(τ) of g(τ)-parallel tensors on M [0,T] via
×
D V =0 and V(0)=V .
τ 0
3. Ka¨hlerity near spatial infinity
NowweassembletheproofofthefirstassertioninTheorem1. Thestrategyhere
is fundamentally the same as in [KW]: from the asymptotically conical soliton, we
constructa self-similar solutionto the backwardRicci flowdefined for τ (0,1]on
∈
a sufficiently distant end, which (after adjustment by a suitable diffeomorphism)
convergessmoothly asτ 0 to the conical(Ka¨hler)metric onsome neighborhood
ց
of infinity. This transforms what is, on its face, an essentially elliptic problem of
uniquecontinuationatinfinityintoaparabolicproblemofbackwarduniquenesson
a finite time interval.
As in the introduction, let (Σ,g ) denote a compact Riemannian manifold of
Σ
dimension n 1, and let g = dr2 + r2g . Denote by r : E R the radial
c Σ c 0
− →
distance r (r,σ)=r relative to g .
c c
Proposition 8 (Proposition 2.1, [KW]). Suppose (M,g¯,f¯) is a shrinking Ricci
soliton asymptotic to the regular cone (E ,g ) along the end V M. Then there
0 c
⊂
exist K , N , and R >0, and a smooth family of maps Ψ :E V defined for
0 0 0 τ R0 →
τ (0,1] satisfying:
∈
(1) For each τ (0,1], Ψ is a diffeomorphism onto its image and Ψ (E ) is
∈ τ τ R0
an end of V.
(2) Thefamilyofmetricsg(x,τ)+τΨ∗g¯(x)isasolutiontothebackwardsRicci
τ
flow (2) for τ (0,1], and converges smoothly as τ 0 to g(x,0) g (x)
c
∈ ց ≡
on E .
R0
(3) For all m=0,1,2,...,
(19) sup rm+2+1 (m)R K .
ER0×[0,1](cid:0) c (cid:1)(cid:12)(cid:12)∇ (cid:12)(cid:12)≤ 0
(cid:12) (cid:12)
Here = and = denote the norm and the Levi-Civita
g(τ) g(τ)
|·| |·| ∇ ∇
connection associated to the metric g =g(τ).
(4) If f is the function on E (0,1] defined by f(τ)=Ψ∗f¯, then τf extends
R0× τ
to a smooth function on all of E1 and there g and τf together satisfy
R0
N N
(20) lim4τf(x,τ)=r2(x), r2 0 4τf r2+ 0,
τց0 c c − rc2 ≤ ≤ c rc2
and
∂ g
(21) (τf)=τS, τ2 f 2 τf = τ2S, τRc(g)+τ f = .
∂τ |∇ | − − ∇∇ 2
KA¨HLERITY OF SHRINKERS ASYMPTOTIC TO KA¨HLER CONES 9
Here S denotes the scalar curvature of g.
We apply the above proposition to a shrinker (M,g¯,f¯) asymptotic to a K¨ahler
cone(E ,g )alongthe endV M asinTheorem1,andassumeforthe restofthe
0 c
⊂
section that g = g(τ) is the smooth family of metrics on E [0,1] it provides.
R0 ×
LetJ denotethecomplexstructureassociatedto(E ,g ),andletJ =J(τ)denote
c 0 c
thefamilyofalmost-complexstructuresobtainedassolutionstothefiberwise-ODE
(3). Then define Pˆ, A, B, Rˆ, and Sˆ as above, in terms of the solution g(τ) and let
X(τ)=Rˆ(τ) Sˆ(τ) V (T∗M V),
⊕ ∈ ⊕ ⊗
Y(τ)=A(τ) B(τ) (T∗M V) (T2(T∗M) V)
⊕ ∈ ⊗ ⊕ ⊗
where V =End( 2T∗M). The families X and Y of sections are smoothly defined
∧
on all of E [0,1] and vanish identically on E 0 .
R0 × R0 ×{ }
Combining (19) with Proposition 6, we see that
∂X C
+∆X (X + Y)
(cid:12)∂τ (cid:12)≤ r2 | | | |
(22) (cid:12) (cid:12) c
(cid:12) ∂Y(cid:12) C
(cid:12) (cid:12) C(X + X)+ Y
(cid:12)∂τ (cid:12)≤ | | |∇ | r2| |
(cid:12) (cid:12) c
(cid:12) (cid:12)
for some C = C(n,K0) on(cid:12)ER0(cid:12)× [0,1]. Here | · | = | · |g(τ), ∇ = ∇g(τ), and
∆=∆ . Moreover,we also have
g(τ)
sup X + X + Y C(n,K ).
0
{| | |∇ | | |}≤
ER0×[0,1]
The boundedness of the components of X follows directly from (19). For the re-
maining components, note first that we can bound ∂ Pˆ via (19), and hence also
∂τ∇
A = Pˆ. With (19), we can then bound the components of X. Similarly, one
∇ ∇
can estimate B= Pˆ via the equation for ∂ Pˆ and the bound on Pˆ.
∇∇ ∂τ∇∇ ∇
The vanishing of X and Y near infinity is now a consequence of the Carleman
estimates established in Propositions 4.7, 5.7, and 5.9 of [KW].
Theorem 9 ([KW]). Suppose (M,g(τ)) is a self-similar solution to (2) on E
R0 ×
(0,1] with potential f which satisfies the conclusions of Proposition 8 relative to the
conical metric g and the parameters K and N . Let X, Y be smooth, uniformly
c 0 0
bounded families of sections of tensor bundles over E [0,1] with the property
R0 ×
that, for any ǫ>0, there is an R =R (ǫ) R such that
1 1 0
≥
∂X
+∆X ǫ(X + Y)
(cid:12)∂τ (cid:12)≤ | | | |
(23) (cid:12) (cid:12)
(cid:12) ∂Y(cid:12)
(cid:12) (cid:12) C (X + X)+ǫY
(cid:12)∂τ (cid:12)≤ 0 | | |∇ | | |
(cid:12) (cid:12)
(cid:12) (cid:12)
for some constant C0 >0 on(cid:12)ER1(cid:12)×[0,1]. Then, if X and Y vanish identically on
E 0 ,theyvanishidentically onE [0,τ ]for someR R andτ (0,1).
R0×{ } R2× 0 2 ≥ 1 0 ∈
The two sets of Carleman estimates given in Proposition 4.9 and Propositions
5.7 and 5.9 in [KW] are valid on any asymptotically conical shrinking self-similar
familyofbackgroundsolutionsg(τ),andtheargumentinSection6ofthatreference
can be applied directly to our setting to combine them to prove the vanishing of
X and Y. Although the particular components of the system (X,Y) in [KW] –
the same asthatin [K1,K3]– differ fromthoseofthe systemconsideredhere(and
10 BRETTKOTSCHWAR
in fact are sections of different bundles), the structural assumptions in Theorem
9 are the only properties of the system used in the application of the Carleman
inequalities. But for the labeling of some constants, the argument which follows
equations (6.1) and (6.2) in [KW] can be used here without change.
Once we know that Rˆ and Pˆ vanish on E [0,τ ] for some R and τ > 0,
∇ R2 × 0 2 0
we can apply Lemma 7 to conclude that g(τ) is K¨ahler with respect to J on the
c
same set. Fix s (0,τ ] and let W = Ψ (E ), and Φ = Ψ−1 : W E , where
∈ 0 s R2 s → R2
Ψ : E V is the map from Proposition 8. Then g¯ = s−1Φ∗(g(s)) is K¨ahler
s R2 → |W
relative to J =Φ∗J , and Φ is the desired biholomorphism.
c
4. Global Ka¨hlerity in the complete case.
TofinishtheproofofTheorem1,wenowarguethattheK¨ahlerstructuredefined
on the end W M must extend to all of M when (M,g) is complete. The real-
⊂
analyticity of Ricci solitons provides almost all that we need.
Lemma10. Suppose(M,g)isacompleteconnectedreal-analyticmanifold, E M
⊂
is open and connected, and p E. If π (E,p) ֒ π (M,p) is surjective, then any
1 1
∈ →
parallel complex structure J on E may be extended uniquely to a parallel complex
E
structure J on M.
Proof. Let Jˆ = J (p). The uniqueness of any extension J of J to M is clear,
E E
since, given q M we must have
∈
(24) J(q)=P JˆP−1
γ γ
where P : T M T M is parallel translation along any piecewise smooth path
γ p q
→
γ from p to q. On the other hand, we can also use (24) to define J(q) : T M
q
→
T M. This willyield asmoothparallelcomplex structure onallofM providedthe
q
extension can be shown to be independent of the path γ.
Suppose σ is any other path from p to q. Under our assumption on π (E), the
1
loopγ σ¯, where σ¯ denotes the reverseparametrizationof σ fromq to p along σ, is
·
homotopic to some loop α contained entirely in E. Then
P =P P P P P =P P P
γ σ σ¯ γ α¯ α σ β α
where β =α¯ γ σ¯ is a null-homotopic loop based at p.
· ·
Since (M,g) is real-analytic, the local holonomy of g at p coincides with the
reduced holonomy (see, e.g., [No]), so both Hol0(M,g) and Hol (E,g) leave Jˆ
p p
invariant. Therefore P JˆP−1 = P JˆP−1 = Jˆ, and so P JˆP−1 = P JˆP−1 as
β β α α γ γ σ σ
desired. Of course, the restriction of J to any path contained in E must coincide
E
withtheparalleltranslationofJˆalongthatpath,sotheconnectednessofE implies
that J =J . (cid:3)
E E
|
WenowgiveanadhocargumenttoshowthattheK¨ahlerstructureextendstothe
entire manifold. The key ingredient is the result of Munteanu-Wang [MW1] which
guarantees that any complete K¨ahler shrinking gradient Ricci soliton is connected
atinfinity. WeapplythistotheuniversalcoverofM todeterminethatthepreimage
of any end in M is connected in the universal cover.
Lemma11. Suppose(M,g,f)isacompletenoncompactgradientRiccisoliton and
g is K¨ahler on some end E M with complex structure J . Then J extends
E E E
| ⊂
uniquely to a complex structure J on M relative to which g is K¨ahler.