Positivity of Ricci curvature under the Kähler--Ricci flow

0.15 MB·English

by  Dan Knopf

POSITIVITY OF RICCI CURVATURE UNDER THE
KA¨HLER–RICCI FLOW
5
0
0
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n
a
J

0
1. Introduction
1

In analyzing a geometric evolution equation, it is of critical importance to de-
]
G termine which curvature conditions are preservedor attained under the flow. This
is especially true for the Ricci and K¨ahler–Ricciflows, because of the fundamental
D
waysinwhichtheseflowscanexploitcurvatureconditionstorevealtopologicaland
.
h canonical geometric properties of their underlying manifolds.
t
a Aninvariant cone inthespaceofcurvatureoperatorsisonethatispreservedby
m
aflow. ForRicciflow,the conditionR 0 ispreservedin alldimensions,while the
≥
[ conditionR 0ispreservedonlyinrealdimensiontwo. Positivecurvatureoperator
≤
is preservedin alldimensions[11],but positive sectionalcurvature is notpreserved
2
v indimensionsfourandabove. Theknowncounterexamples,constructedrecentlyby
8 Ni [18], are non-product metrics on the total space of the tangent bundles Rn ֒
0 T n ։ n for n 2. However, positive sectional curvature and positive Ric→ci
1 S S ≥
curvaturearepreservedindimensionthree[10]. Inhigherdimensions,Huisken[14]
1
and Nishikawa [19, 20] have demonstrated the invariance of certain sets defined by
0
5 curvaturepinchingconditions. Margerin[16]improvedHuisken’sconstant,thereby
0 establishingasharpinvariantsetindimensionfour. Alsoindimensionfour,H.Chen
/ proved that 2-positivity of the curvature operator is preserved [5], while Hamilton
h
t later proved that positive isotropic curvature constitutes an invariant cone [13].
a
An attractive cone in the space of curvature operators is one that is entered
m
asymptotically at a finite-time singularity. For example, the estimate R R (0)
min
: ≥
v implies that the half-space R 0 constitutes an attractive cone in all dimensions.
≥
i Attractive cones place valuable restrictions on which singularity models may ap-
X
pear. Important examples are the pinching estimates for 3-manifolds proved inde-
r
pendently by Ivey [15]andHamilton[12]. These estimatesimply thatany rescaled
a
limit formed at a finite-time singularity in dimension three has nonnegative sec-
tionalcurvature. In turn, this fact is of fundamental importance in Perelman’s use
ofκ-solutions[21,22]inhis recentspectacularprogresstowardthe Geometrization
Conjecture.
For K¨ahler–Ricci flow, Bando [1] and Mok [17] proved that positive bisectional
curvaturedefines aninvariantcone. PositiveRiccicurvature is a moreproblematic
condition (except of course in complex dimension one). It is not expected to be
an attractive cone. Indeed, certain K¨ahler–Ricci solitons constructed by Feldman,
Ilmanen, and the author [9] have Ricci curvature of mixed sign, yet are strongly
Date:February1,2008.
PartiallysupportedbyNSFgrantDMS-0328233.
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conjectured to occur as rescaled limits of singularities encountered by perturbed
metrics on the Calabi manifolds CP1 ֒ Fk ։CPn−1. (See Example 2.2 of [9].)
→
On the other hand, it would be very useful indeed if positive Ricci curvature
constituted an invariant cone for K¨ahler–Ricci flow. For example, parts of the
proofs of some recent important results of X.-X. Chen and Tian for K¨ahler–Ricci
flow [6, 7] only require positive Ricci curvature, but the authors impose the much
strongerassumptionof positive bisectionalcurvature to be sure that positive Ricci
curvature will be preserved. (These papers also assume the existence of a K¨ahler–
Einsteinmetricontheunderlyingmanifold. SeeinparticularRemarks1.5–1.7and
Question9.3of[7].) While positiveRiccicurvatureis notexpectedto be preserved
in high complex dimensions, there were until now promising reasons to hope that
it might be preserved in complex dimension two. Unpublished work of Cao and
⊥
Hamilton proves that positive orthogonal bisectional curvature B(x,x ) 0 is
≥
preservedunder K¨ahler–RicciflowandthatitspresenceensuresthatpositiveRicci
curvature is also preserved [4]. Very recent results of Phong and Sturm prove
that positive Ricci curvature is preserved on a compact complex surface with the
propertythat the sum of any two eigenvaluesof its tracelesscurvature operatoron
traceless (1,1)-forms is non-negative [23]. (See also [24].)
ThemainpurposeofthisshortnoteistoshowthatpositiveRiccicurvatureisnot
preservedfor arbitrarycomplex surfaces. We constructcomplete K¨ahlermetrics of
boundedcurvatureandnon-negativeRiccicurvaturewhoseK¨ahler–Riccievolutions
immediately acquire Ricci curvature of mixed sign. Let L2 denote C2 blown up
−1
atthe origin. (See below foranexplanationofthe notation.) Inthis paper,wewill
prove the following:
Theorem 1. There exists a one-parameter family (modulo scaling) of Ka¨hler met-
rics on L2−1 that are complete, of bounded curvature, and of non-negative Ricci
curvature, but which immediately develop mixed Ricci curvature when evolved by
the Ka¨hler–Ricci flow.
Infact,Theorem1isaspecialcaseofamoregeneralresult. Forn 2andk N,
let Ln denote the total space of the holomorphic line bundle C֒ ≥Ln ։CP∈n−1
−k → −k
characterized by the equation c ,[Σ] = k, where c is the first Chern class of
1 1
the bundle and Σ CP1 is a phositiveliy ori−ented generator of H (CPn−1;Z). Since
2
the complement of≈the zero-section of Ln is biholomorphic to (Cn 0 )/Z , one
mayconstructLn bygluingaCPn−1 on−tok(Cn 0 )/Z attheorigi\n{. N}otickethat
Ln is simply th−ektautological line bundle, i.e. C\{n b}lownk up at the origin.
−1
Theorem 2. For each complex dimension n 2 and all integers k [1,n 1],
≥ ∈ −
there exists a one-parameter family (modulo scaling) of Ka¨hler metrics on Ln that
−k
are complete, of bounded curvature, and of non-negative Ricci curvature, but which
immediately develop mixed Ricci curvature when evolved by the Ka¨hler–Ricci flow.
Theremainderofthispaperisorganizedasfollows. Sincethe criticaln=2case
is most interesting, we begin with a detailed proof of Theorem 1. In Section 2 we
provide, for the convenience of the reader, a review of U(2)-invariant metrics on
C2 0 . In Section 3, we construct initial metrics on L2 . Then in Section 4, we
\{ } −1
finishtheproofofTheorem1. Finally,inSection5,weindicatethestraightforward
generalizations by which one proves Theorem 2 for all complex dimensions n 2.
≥
POSITIVITY OF RICCI CURVATURE 3
Acknowledgement1. TheauthorwarmlythanksHuai-DongCaoforencouraging
him to work on this problem.
Remark 1. Like Ni’s counterexamples for non-negative sectional curvature [18],
the manifolds constructed here are noncompact. Thus it is still an open question
of significant interest to find compact counterexamples for which non-negative Ricci
(or sectional) curvatures are not preserved under Ka¨hler–Ricci flow.
2. U(2)-invariant Ka¨hler metrics
In this section, we review the construction of U(2)-invariant K¨ahler metrics on
C2 0 . Such metrics also appear in [2, 3] and [9]. Consider C2 0 with complex
\{ } \{ }
coordinates (z ,z ). Define
1 2
u= z 2, v = z 2, and w =u+v.
1 2
| | | |
Anyfunction P :C2 0 R depending onlyonw may be writtenasP(r), where
\{ }→
r =logw.
Anticipating thatwe willlaterabuse notationby regardingP asafunction oftime
as well, we shall use subscripts to denote (partial) derivatives with respect to r.
Define
(2.1) ϕ(r)=P (r).
r
Then P is the K¨ahler potential for a metric g if and only if the derivatives ϕ and
ϕ are everywhere positive. In this case, it is straightforwardto compute that
r
(2.2) g =e−rϕδ +e−2r(ϕ ϕ)z¯z
i¯ ij r i j
−
and
(2.3) gi¯=erϕ−1δij +(ϕ−1 ϕ−1)ziz¯j.
r −
Withrespecttothestandardbases(∂/∂z ,∂/∂z )and(dz ,dz ),onecantherefore
1 2 1 2
regardg and g−1 as the matrices
g1¯1 g1¯2 = 1 vϕ+uϕr (ϕr −ϕ)z¯1z2
(cid:18)g2¯1 g2¯2(cid:19) w2 (cid:18)(ϕr−ϕ)z1z¯2 uϕ+vϕr (cid:19)
and
g1¯1 g2¯1 1 uϕ+vϕ (ϕ ϕ )z¯ z
(cid:18)g1¯2 g2¯2(cid:19)= ϕϕr (cid:18)(ϕ−ϕr)z1rz¯2 v−ϕ+ruϕ1r 2(cid:19)
respectively. In the standard coordinate chart, define
(2.4) G= logdetg =2r logϕ logϕ
r
− − −
and
ϕ ϕ
r rr
(2.5) ψ =G =2 .
r
− ϕ − ϕ
r
Then the complex Ricci tensor Rc = R dz dz¯ may be represented in the same
i¯ i j
basis as
(2.6) R1¯1 R1¯2 = 1 vψ+uψr (ψr−ψ)z¯1z2 .
(cid:18)R2¯1 R2¯2(cid:19) w2 (cid:18)(ψr −ψ)z1z¯2 uψ+vψr (cid:19)
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Abusingterminologyslightly,wewillsayV isaneigenvectorofRiccicorresponding
to the eigenvalue λ if R Vi =λg Vi. Thus understood, the eigenvalues of Rc are
i¯ i¯
ψ ψ
r
λ = and λ = .
1 2
w w
The Levi-Civita connection of g is determined by the Christoffel symbols
z¯ ϕ ϕ
Γ1 = 1 u rr +2v r +u 2w
11 w2 (cid:20) ϕ ϕ − (cid:21)
r
z¯ ϕ ϕ
Γ1 = 2 u rr +(v u) r v
12 w2 (cid:20) ϕ − ϕ − (cid:21)
r
z z¯2 ϕ ϕ
Γ1 = 1 2 rr 2 r +1
22 w2 (cid:20) ϕ − ϕ (cid:21)
r
z¯2z ϕ ϕ
Γ2 = 1 2 rr 2 r +1
11 w2 (cid:20) ϕ − ϕ (cid:21)
r
z¯ ϕ ϕ
Γ2 = 1 v rr +(u v) r u
12 w2 (cid:20) ϕ − ϕ − (cid:21)
r
z¯ ϕ ϕ
Γ2 = 2 v rr +2u r +v 2w
22 w2 (cid:20) ϕ ϕ − (cid:21)
r
and their conjugates. The complex Riemann tensor is determined by
Ri¯kℓ¯=−gmℓ¯∂¯jΓmik =−∂i∂¯jgkℓ¯+gpq¯∂igkq¯∂¯jgpℓ¯.
A straightforwardcomputation shows that
ϕ2 ϕ2
Ri¯kℓ¯=e−4r(cid:20)−ϕrrr+4ϕrr−2ϕr+2ϕ−4 ϕr + ϕrr(cid:21)z¯izjz¯kzℓ
r
ϕ2
+e−3r ϕ +ϕ ϕ+ r (z¯z δ +z¯δ z +δ z¯ z +δ z z¯ )
rr r i j kℓ i jk ℓ ij k ℓ iℓ j k
(cid:20)− − ϕ (cid:21)
+e−2r[ ϕ +ϕ](δ δ +δ δ ).
r ij kℓ iℓ jk
−
Toputthisintoamoreusefulform,wefollow[2]. SincethemetricisU(2)-invariant,
there is no loss of generality in evaluating the curvature at (ζ,0), where ζ = 0 is
6
arbitrary. Furthermore, one may assume that the complex tangent space at (ζ,0)
is spanned by an orthonormal basis of complex vectors x,y such that x2 = 0.
Then one can readily compute the bisectional curvatures B(x,x) = R(x,x¯,x,x¯),
B(x,y)=R(x,x¯,y,y¯), and B(y,y)=R(y,y¯,y,y¯) at (ζ,0), obtaining
1 ϕ2
B(x,x) = ( rr ϕ )x1 4
|(ζ,0) ζ 4 ϕ − rrr | |
r
| |
1 ϕ2 ϕ2
B(x,y) = ( rr ϕ )x1 2 y1 2+( r ϕ )x1 2 y2 2
|(ζ,0) ζ 4 (cid:20) ϕ − rrr | | | | ϕ − rr | | | | (cid:21)
r
| |
1 ϕ2 ϕ2
B(y,y) = ( rr ϕ )y1 4+4( r ϕ )y1 2 y2 2+2(ϕ ϕ )y2 4 .
|(ζ,0) ζ 4 (cid:20) ϕ − rrr | | ϕ − rr | | | | − r | | (cid:21)
r
| |
Observe in particular that the choices
ζ ∂ ζ ∂
(2.7) X = | | and Y = | |
|(ζ,0) √ϕr ∂z1 |(ζ,0) √ϕ ∂z2
POSITIVITY OF RICCI CURVATURE 5
allow one to recover (2.6) in the form
ψ
Rc(X,X¯) =B(X,X)+B(X,Y)= r
|(ζ,0) ϕ
r
ψ
Rc(Y,Y¯) =B(Y,Y)+B(X,Y)= .
|(ζ,0) ϕ
3. An initial metric of positive Ricci curvature
We now wantto investigatewhether there exists a U(2)-invariantK¨ahlerpoten-
tial P on C2 0 with the following global properties:
\{ }
(1) ϕ>0 everywhere;
(2) ϕ >0 everywhere;
r
(3) ψ >0 everywhere;
(4) ψ 0 everywhere;
r
≥
(5) g has bounded curvature;
(6) g extends smoothly to a complete metric as r + ; and
→ ∞
(7) g extends smoothly to a complete metric as r .
→−∞
Furthermore,werequirethatthereexistsatleastonepoint(ζ,0) C2 0 such
∈ \{ }
that with respect to an orthonormal basis x,y for the complex tangent space at
(ζ,0), the following local properties hold:
I B(x,y)<0; and
II B(x,x)= B(x,y).
−
It may seem foolhardy to expect a single function P to satisfy all nine of these
requirements. However, let a>0 be a constant to be determined and consider the
second-order ode
(3.1) [log(ϕϕ )] =a.
r r
Its generalsolution is 2ear+b/a+c, where b and c are arbitrary. Without loss of
generality, one may elipminate b by scaling, yielding
2
(3.2) ϕ(r)= ear+c.
ra
One then has
(3.3) ϕϕ =ear,
r
so that global properties (1) and (2) are satisfied whenever c 0. Moreover,
≥
ψ =2 a
−
everywhere. So properties (3) and (4) are satisfied whenever a (0,2).
∈
To evaluate the bisectional curvatures of the resulting metric, it suffices to con-
sider the orthonormal basis (2.7) for the complex tangent space at (ζ,0), where
ζ =0 is arbitrary. One finds that
6
ac
(3.4) B(X,X)= = B(X,Y),
ϕ3 −
so that local properties I and II will in fact be satisfied globally if c > 0. The
choices a (0,2) and c>0 also ensure that
∈
2 2
B(Y,Y)= 1 ear+c >B(X,X).
ϕ3 (cid:20)(cid:18)a − (cid:19) (cid:21)
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Because ϕ(r) √c > 0 as r and ϕ(r) = (e(a/2)r) as r , the cur-
→ → −∞ O → ∞
vatures are globally bounded, satisfying property (5). To investigate completeness
at spatial infinity, observe that (3.3) implies that ϕ = (ear) as r and
r
O → −∞
that ϕ = (e(a/2)r) as r . Since radial paths emanating from the origin are
r
O → ∞ ∞
geodesics and the length of such a path is −∞√ϕrdr, it is easy to see that the
metric is complete as r + , satisfying glRobal property (6).
→ ∞
On the other hand, the metric distance from an arbitrary point to r = is
−∞
finite. Our choice ofc>0 makes it impossible to complete the metric smoothly by
gluing in a point at the origin. But Calabi’s theorem [8] tells us that g will extend
to a smoothK¨ahler metric on L2 if ϕ has anasymptotic expansionnear w=0 of
−k
the form
ϕ=a +a wk+a w2k+ (w3k),
0 1 2
O
witha anda bothpositive. (SeeSection4.1of[9]foranexposition.) Thefunction
0 1
(3.2)willhavesuchanexpansionifaisapositiveinteger. Fortunatelyforus,there
isa (unique)positiveintegerinthe interval(0,2). Takinga=1 thuslets us satisfy
global property (7). Notice that for each c > 0, the solution ϕ then admits the
expansion
1 1
ϕ=√c+ w w2+ .
√c − 2c3/2 ···
In particular, 2π√c is the area of the CP1 that constitutes the zero-section of the
bundle.
Remark 2. The Ka¨hler manifolds (L2−1,g) we have constructed are asymptotic at
infinity to a Ka¨hler cone Γ on C2 0 whose cone angle around the origin in each
\{ }
complex line is π and whose density ratio is VolΓ(B1)/VolC2(B1)=1/4.
4. Effect of the Ka¨hler–Ricci flow
We now consider the K¨ahler–Ricci evolution (L2 ,g(t)), where we take P(r,0)
−1
tobetheK¨ahlerpotentialP(r)constructedabove. Againchooseanarbitrarypoint
(ζ,0) C2 0 . Because
∈ \{ }
1 ϕ 0
g = r
|(ζ,0) ζ 2 (cid:18)0 ϕ(cid:19)
| |
and
1 ψ 0
Rc = r
|(ζ,0) ζ 2 (cid:18)0 ψ(cid:19)
| |
in the standard basis, the K¨ahler–Ricci flow
∂
g = Rc
∂t −
will be satisfied if and only if ϕ evolves by the pde ϕ = ψ; to wit,
t
−
ϕ ϕ
rr r
(4.1) ϕ = + 2.
t
ϕ ϕ −
r
Since ∂/∂r and∂/∂tcommute,itfollowsthatthe Riccipotentialψ mustevolveby
the pde
ψ ϕ ψ ψ ϕ ψ
rr rr r r r
(4.2) ψ = + .
t ϕ − ϕ2 ϕ − ϕ2
r r
POSITIVITY OF RICCI CURVATURE 7
Since ψ(,0) 1, this reduces to
· ≡
∂ ϕ
r
ψ =
∂t (cid:12) −ϕ2
(cid:12)t=0
(cid:12)
at the initial time. A further computa(cid:12)tion then shows that
∂ er
(4.3) ψ = (er c),
∂t r(cid:12) ϕ5 −
(cid:12)t=0
(cid:12)
which is strictly negative for all r(cid:12)< logc. Since ψ (,0) 0, the complex Ricci
r
· ≡
tensor must acquire a negative eigenvalue λ <0 at all such points for small times
2
t>0. This completes the proof of Theorem 1.
5. The general case: complex dimension n 2
≥
If P is a U(n)-invariant K¨ahler potential on Cn 0 , formulas (2.2) for g and
(2.3) for g−1 remain unchanged. Formula (2.4) beco\m{e}s
G=nr logϕ logϕ ,
r
− −
whence (2.5) is replaced by
ϕ ϕ
r rr
ψ =G =n .
r
− ϕ − ϕ
r
One calculates easily that
R =e−rψδ +e−2r(ψ ψ)z¯z .
i¯ ij r i j
−
To display the eigenvalues of Rc, one may without loss of generality fix Z =
(ζ,0,...,0) Cn 0 with ζ =0 arbitrary. Then
∈ \{ } 6
ϕ
r
1  ϕ 
g|Z = ζ 2  ... 
| |  
 ϕ
 
and
ψ
r
1  ψ 
Rc|Z = ζ 2  ... .
| |  
 ψ
 
If ϕ is defined by (3.2) with a>0, one finds that
ψ n a,
≡ −
hence that ψ 0. Equation (4.1) becomes
r
≡
ϕ ϕ
rr r
ϕ = + n,
t
ϕ ϕ −
r
so that (4.2) remains unchanged. The analog of (4.3) is
∂ ear
ψ =(n a) (ear ac),
∂t r(cid:12) − ϕ5 −
(cid:12)t=0
(cid:12)
(cid:12)
8 DANKNOPF
which is strictly negative for all a (0,n) and c>0 whenever r < 1 log(ac). If we
∈ a
takea (0,n)to be aninteger,then ϕ willhave anasymptoticexpansionnearthe
∈
origin of the form
1 1
ϕ=√c+ wa w2a+ .
a√c − 2a2c3/2 ···
Henceby Calabi’stheorem,g willextendtoa smoothmetric onLn−a for allsucha.
It is easy to check that g is complete and asymptotic to a K¨ahler cone at infinity,
hence of bounded curvature. This completes the proof of Theorem 2.
Remark 3. For each 1 k n 1, the Ka¨hler manifolds (Ln ,g) we have
constructed are asymptotic≤at in≤finity−to a Ka¨hler cone Γ on Cn −0k whose cone
k
\{ }
angle around the origin is π and whose density ratio is VolΓk(B1)/VolCn(B1) =
k1−n/2n.
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University of Texas atAustin
E-mail address: [email protected]
URL:http://www.ma.utexas.edu/~danknopf/