A second variation formula for Perelman's $\mathcal{W}$-functional along the modified Kähler-Ricci flow

0.19 MB·English

by  Nefton Pali

A second variation formula for Perelman’s
W-functional along the modified K¨ahler-Ricci
2
1
flow
0
2

n
a
J NEFTON PALI

4
January 5, 2012

]
G
D
Abstract
.
h We show a quite simple second variation formula for Perelman’s W-
t
a functional along themodified K¨ahler-Ricci flow over Fano manifolds.
m
[
1 Introduction

1
v In a celebrated preprint Grigory Perelman [Per] introduced an entropy type
0
functional denoted by W which gradient flow coincides with Perelman’s modi-
7
fied Ricci flow. In the Fano case this flow coincides with Perelman’s modified
9
0 K¨ahler-Ricci flow. The total second variation of this functional plays an im-
. portant role in the study of the stability and convergence of the K¨ahler-Ricci
1
0 flow over Fano manifolds. Total second variation formulas at a shrinking Ricci
2 solitonpointwereobtainedindependently byCao-Hamilton-Ilmanenen[C-H-I],
1 Cao-Zhu [Ca-Zhu] and Tian-Zhu [Ti-Zhu]. The work of Cao-Zhu [Ca-Zhu] is
:
v based on the previous work of Cao-Hamilton-Ilmanenen [C-H-I]. Important
i applications to the stability and the convergence of the K¨ahler-Ricci flow over
X
FanomanifoldsweregivenbyTian-Zhu[Ti-Zhu]andbyTian-Zhang-Zhang-Zhu
r
[T-Z-Z-Z]. Total second variation formulas at arbitrary points in the space of
a
metricscanbe foundin[Pal]. The formulasin[Pal]areofdifferentnaturefrom
the ones obtained by [C-H-I], [Ca-Zhu] and [Ti-Zhu].
Inthispaperweshowaverysimple secondvariationformulaforPerelman’s
W-functional along the modified K¨ahler-Ricci flow over Fano manifolds.
Our computation concern the K¨ahler-Ricci flow but we feel that the second
variation formula is more meaningful for the modified K¨ahler-Ricci flow. In
any case one can switch easily from one formula relative to a flow to the other
thanks to the invariance by diffeomorphisms of the functional W. It is some
how surprising that we could not prove a simple second variation formula by
working directly with the modified K¨ahler-Ricci flow.
1
Thekeycomputationwhichallowsustoobtainoursecondvariationformula
is based on a very surprising integration by parts formula along the K¨ahler-
Ricci flow. It turns out that the corresponding version of this formula along
the modified K¨ahler-Ricciflow can be obtained using Perelman’s versionof the
twicecontractedBianchiidentity. Thedetails aboutthis lastfactareexplained
in section 3.
We expect convexity of the functional W along the K¨ahler-Ricci flow. Our
formula should be considered as a first step in this direction. It should be clear
for the experts that convexity along the flow implies quite natural and strong
convergence results. We explain in detail below the set-up and our result.
In this paper we will adopt the sign convention ∆ := div∇ in order to not
generate confusion in some standard formulas along the Ka¨hler-Ricci flow.
Let (X,g) be a compact oriented (for simplicity) Riemannian manifold of
dimension 2n and f a smooth real valued function over X. We remind that
Perelman’s W-functional [Per] is defined (up to a constant) by the formula
W(g,f) := |∇ f|2 + Scal + 2f − 2n e−fdV .
g g g g
ZX(cid:16) (cid:17)
It rewrites also as
W(g,f) = ∆ f + Scal + 2f − 2n e−fdV
g g g
ZX(cid:16) (cid:17)
= 2H(g,f)e−fdV .
g
Z
X
where
2H(g,f) := 2∆ f − |∇ f|2 + Scal + 2f − 2n.
g g g g
(We use here the standardidentity ∆ e−f =(|∇ f|2−∆ f)e−f .) The impor-
g g g g
tance of the therm H is known from Perelman’s work [Per] (see the subsection
4.1 in the appendix) and it will appear also in our second variation formula.
Let (X,J ) be a Fano manifold and let (gˆ ) be the K¨ahler-Ricci flow
0 t t>0
d
gˆ = − Ric(gˆ) + gˆ,
t t t
dt
withassociatedsymplecticformωˆ :=gˆJ ∈2πc (X). Consideralsoasolution
t t 0 1
(fˆ) of Perelman’s backwardheat equation
t t∈[0,T]
d
2 fˆ = − ∆ fˆ + |∇ fˆ|2 − Scal + 2n, (1)
dt t gˆt t gˆt t gˆt gˆt
and set Ω := e−fˆ0dV . Consider now the flow of diffeomorphisms (Ψ )
gˆ0 t t∈[0,T]
given by the equation
d
2 Ψ = − ∇ fˆ ◦Ψ , (2)
dt t gˆt t t
(cid:16) (cid:17)
2
and set g := Ψ∗gˆ,J := Ψ∗J ,ω := Ψ∗ωˆ ,f := fˆ ◦ Ψ . Then hold the
t t t t t 0 t t t t t t
evolution formulas
d
g˙ := g = − Ric(g ) − ∇ df + g ,
t dt t t gt t t
d
ω˙ := ω = − Ric (ω ) − i∂ ∂ f + ω ,
t dt t Jt t Jt Jt t t
and
d
2 f = −∆ f − Scal(g ) + 2n.
dt t gt t t
This last rewrites as
d d
2 f = Tr ω ,
dt t ωt dt t
whichis equivalenttothe volumeformpreservingconditione−ftdVgt =Ω. The
flowofK¨ahlerstructures(X,J ,ω ) iscalledPerelman’smodified K¨ahler-
t t t∈[0,T]
Ricciflow. WesetW :=W(g ,f )alongthemodifiedK¨ahler-Ricciflowandwe
t t t
remind Perelman’s [Per] fundamental identity
W˙ = |g˙ |2Ω. (3)
t Z t t
X
Indeed this is equivalent to Perelman’s monotony [Per] of the W-functional
along the K¨ahler-Ricciflow thanks to the invariance of the W-functional under
the action of diffeomorphisms. In our case it gives W = W(gˆ,fˆ). With this
t t t
notations our result states as follows.
Theorem 1. Along the modified K¨ahler-Ricci flow (X,J ,ω ) hold the
t t t∈[0,T]
second variation formula
W¨ = 2 ω˙ ·Rm ,ω˙ + 2 ∇ H 2 − ∇ g˙ 2 Ω,
t ZXh (cid:10) t gt t(cid:11)t (cid:12) t t(cid:12)t (cid:12) t t(cid:12)ti
(cid:12) (cid:12) (cid:12) (cid:12)
whereRm ∈C∞(X,End(Λ2T ))denotestheRiemanncurvatureoperatorand
gt X
H :=H(g ,f ).
t t t
2 The second variation of W along the
K¨ahler-Ricci flow
All the computations in this section are done with respect to the K¨ahler-Ricci
flow (X,J ,gˆ). From now on all the complex operators will depend on the
0 t
complex structure J . Therefore we will suppress the dependence on J when
0 0
itisredundant. Moreoverallthe covariantderivativesandnormsarecomputed
withrespectto the K¨ahler-Ricciflow. Thus fornotationsimplicity wewilldrop
3
thedependenceontheevolvingmetricandonthetime. AlongtheK¨ahler-Ricci
flow we define the heat operator
∂
(cid:3) := ∆ − 2
∂t
and the conjugate heat operator
∂
(cid:3)∗ := ∆ + 2 − Scal + 2n.
∂t
The terminology is justified by the formula
d
2 (ab)dV = − ((cid:3)ab − a(cid:3)∗b)dV ,
gˆ gˆ
dtZ Z
X X
for any a,b ∈ C∞(X × R ,R). We observe that with this notations the
≥0
backwardheat equation (1) is equivalent to the equation
(cid:3)∗e−fˆ = 0. (4)
For any α ∈ Λ2T∗ we define the endomorphism α∗ := ωˆ−1α. Let now for
X
notation simplicRity
Ric := Ric (ωˆ ),
J0 t
Ric := Ric + i∂∂fˆ,
α := ωˆ − Ric,
A := ∂∇fˆ.
Thus time deriving Perelman’s monotony formula for the W-functional along
the K¨ahler-Ricci flow
W˙ = |α∗|2 + |A|2 e−fˆdV ,
gˆ
Z
X(cid:0) (cid:1)
(see formula (20) in the subsection 4.1 of the appendix) we obtain
W¨ = − 1 (cid:3) |α∗|2 + |A|2 e−fˆdV . (5)
gˆ
2 Z
X (cid:0) (cid:1)
In the next subsections we will expand the integrand therm.
2.1 Computation of the time variation of the squared
norms
We introduce first a notation. Let E :=End (T ). For all ξ ∈T we define
C X,J0 X
the J -anti-linear operator ∂ Ric∗·ξ as
0 E
∂ Ric∗·ξ (η):= ∂ Ric∗ (ξ), ∀η ∈T .
E E,η X
(cid:0) (cid:1) (cid:0) (cid:1)
4
The evolution equation of the Ricci form Ric and the evolution equation (1) of
fˆimply the identity
d d
2 Ric = i∂∂¯ Scal + 2 fˆ = i∂∂ |∇fˆ|2 − ∆fˆ .
dt (cid:18) dt (cid:19) (cid:16) (cid:17)
Moreover
d d d
Ric = ωˆ Ric∗ + ωˆ Ric∗
dt (cid:18)dt (cid:19) dt
d
= Ric − Ric Ric∗ + ωˆ Ric∗ ,
dt
and thus
∗
d d
Ric∗ = Ric + Ric∗ Ric∗ − Ric∗ .
dt (cid:18)dt (cid:19)
The fact that the endomorphism α∗ = I − Ric∗ is gˆ-symmetric implies the
identity |α∗|2 =Tr (α∗)2. Time deriving this identity we infer
R
d d d
|α∗|2 = − Tr Ric∗α∗ + α∗ Ric∗
dt R(cid:18)dt dt (cid:19)
d
= − 2Tr α∗ Ric∗
R(cid:18) dt (cid:19)
∗
d
= − 2 α∗, Ric + Ric∗ Ric∗ − Ric∗ .
(cid:28) (cid:18)dt (cid:19) (cid:29)
Inordertoobtainthethirdequalityweneedtoobservetheelementaryidentities
hRic∗ Ric∗,α∗i = Tr (Ric∗Ric∗α∗)
R
= Tr (α∗Ric∗ Ric∗)
R
= hα∗,Ric∗Ric∗i . (6)
In conclusion hold the identity
d ∗
|α∗|2 = α∗, i∂∂(∆fˆ − |∇fˆ|2) − 2 Ric∗ Ric∗ + 2Ric . (7)
dt
D h i E
We compute now the time variation of the operator A. Time deriving the
definition of the gradient of fˆwe infer
d d d
d fˆ = ∇fˆ ¬gˆ + ∇fˆ ¬ gˆ
t
(cid:18)dt (cid:19) (cid:18)dt (cid:19) dt
d
= ∇fˆ ¬gˆ + dfˆ − ∇fˆ¬ Ric(gˆ),
(cid:18)dt (cid:19)
5
and thus using the elementary identity Ric(gˆ)=− Ric·J we obtain
0
d d
∇fˆ = ∇ fˆ − fˆ + Ric∗·∇fˆ.
dt (cid:18)dt (cid:19)
We deduce the expression
d d
A˙ := A = ∂ ∇ fˆ − fˆ + ∂ Ric∗·∇fˆ + Ric∗A.
dt (cid:18)dt (cid:19) E
On the other hand |A|2 = Tr A2 since A is also gˆ-symmetric. Time deriving
R
this identity we infer the equalities
d
|A|2 = Tr A˙A + AA˙
dt R
(cid:16) (cid:17)
d
= 2 A,∂ ∇ fˆ − fˆ + ∂ Ric∗·∇fˆ
(cid:28) (cid:18)dt (cid:19) E (cid:29)
+ 2 A2,Ric∗ .
(cid:10) (cid:11)
In order to obtain the last therm we use the identity
Tr (ARic∗A)= A2,Ric∗ . (8)
R
(cid:10) (cid:11)
We infer from the evolution equation (1) of fˆthe formula
d
|A|2 = A,∂ ∇ |∇fˆ|2 − ∆fˆ − Scal − 2fˆ
dt
D (cid:16) (cid:17)E
+ 2 A,∂ Ric∗·∇fˆ + 2 A2,Ric∗ . (9)
E
D E
(cid:10) (cid:11)
2.2 Computation of the Laplacian of the squared norms
At an arbitrary space time point (x ,t ) ∈ X ×[0,T] we fix J -holomorphic
0 0 0
and gˆ -geodesic coordinates (z ,...,z ) centered at the point x and we set
t0 1 n 0
ζk := ∂∂zk. At the time t0 let ∇r := ∇ζr, ∇r¯ := ∇ζ¯r and set er := ζr + ζ¯r.
Expanding and canceling we obtain the expression
∆ = ∇ ∇ + ∇ ∇ = 2∇ ∇ + 2∇ ∇ . (10)
er er J0er J0er r r¯ r¯ r
Part I. Let
Rk,¯l := Ric(ζk,ζ¯l),
R′ := Ric(ζ ,ζ¯).
k,¯l k l
With this notations hold the local expression
Ric∗ = Rk,¯l ζk∗⊗ζl + Rk,¯l ζ¯k∗⊗ζ¯l , Rk,¯l = 2Rk′,p¯ωˆp,¯l ,
6
where
i
ωˆ = 2 ωk,¯l ζk∗∧ζ¯l∗ .
Using the expression (10), the vanishing properties of the complexified Levi-
Civita connection and the previous identities we infer the expressions at the
space time point (x ,t )
0 0
∆Ric∗ = 4∂r2,r¯Rk,¯l ζk∗⊗ζl + 4∂r2,r¯Rk,¯l ζ¯k∗⊗ζ¯l
+ 2Rk,¯l ∇r¯∇r ζk∗⊗ζl + 2Rk,¯l ζk∗⊗∇r¯∇r ζl
+ 2Rl,k¯ ∇r ∇r¯ζ¯k∗⊗ζ¯l + 2Rl,k¯ ζ¯k∗⊗∇r ∇r¯ζ¯l
= 4∂r2,r¯Rk,¯l ζk∗⊗ζl + 4∂r2,r¯Rk,¯l ζ¯k∗⊗ζ¯l
+ 2 Rk,p¯Rp,¯l − Rk,p¯Rp,¯l ζk∗⊗ζl
(cid:0) (cid:1)
+ 2 Rl,p¯Rp,k¯ − Rl,p¯Rp,k¯ ζ¯k∗⊗ζ¯l
(cid:0) (cid:1)
= 4∂r2,r¯Rk,¯l ζk∗⊗ζl + 4∂r2,r¯Rk,¯l ζ¯k∗⊗ζ¯l
+ Ric∗Ric∗ − Ric∗Ric∗ .
At this point we need the following elementary fact.
Lemma 1. At the space time point (x ,t ) hold the identity
0 0
∂r2,r¯Rk,¯l = ∂k2,¯l Rr,r¯ − Rr,p¯Rmkr,,p¯¯l + 2Rk,p¯Rp,¯l ,
where
Rm = Rmr,p¯(ζ∗∧ζ¯∗)⊗(ζ ∧ζ¯ ), Rmr,p¯ = − 2∂2 ωˆ .
k,¯l k l r p k,¯l k,¯l p,r¯
Proof. Spacederivingtwicethe expressionofthe coefficientsRk,¯l andusingthe
K¨ahlersymmetryrelationswefindatthespacetimepoint(x ,t )theexpression
0 0
4∂j2,k¯ Rh,s¯ = − 4∂j4,k¯,h,s¯ωˆl,¯l + Rmls,,r¯h¯Rmlj,,r¯k¯ + Rmlk,,r¯h¯Rmlj,,r¯s¯,
which implies the symmetry of the indices k,s for ∂2 R . The symmetry of
j,k¯ h,s¯
theindicesj,hfollowsbyconjugation. Moreoveratthespacetimepoint(x ,t )
0 0
hold
∂r2,s¯Rk,¯l = 2∂r2,s¯Rk′,¯l − 2Rk′,p¯∂r2,s¯ωˆp,¯l
= 2∂r2,s¯Rk,¯l + 2∂r4,s¯,k,¯l fˆ − 2Rk′,p¯∂r2,s¯ωˆp,¯l .
7
We infer
∂r2,s¯Rk,¯l = ∂k2,¯l Rr,s¯ + 2Rr,p¯∂k2,¯l ωˆp,s¯ − 2Rk,p¯∂r2,s¯ωˆp,¯l ,
and thus the required conclusion.
From lemma 1 we deduce the identity along the K¨ahler-Ricci flow
∆Ric∗ = i∂∂Tr Ric∗ − 2RicRm ∗ + Ric∗Ric∗ + Ric∗ Ric∗.
R
(cid:0) (cid:1) (11)
Part II. We remind first the local expression
A = Ak,¯l ζ¯l∗⊗ζk + Ak,¯l ζl∗⊗ζ¯k ,
Ak,¯l = 2ωˆp,k¯ ∂¯l2,p¯fˆ − ∂¯l ωˆj,p¯ωˆr,¯j ∂r¯fˆ .
h i
(See the subsection 4.2 in the appendix.) As in part I we infer the expressions
at the space time point (x ,t )
0 0
∆A = 4∂r2,r¯Ak,¯l ζ¯l∗⊗ζk + 4∂r2,r¯Ak,¯l ζl∗⊗ζ¯k
+ 2Ak,¯l ∇r ∇r¯ζ¯l∗⊗ζk + 2Ak,¯l ζl∗⊗∇r ∇r¯ζ¯k
+ 2Ak,¯l ζ¯l∗⊗∇r¯∇r ζk + 2Ak,¯l ∇r¯∇r ζl∗⊗ζ¯k
= 4∂r2,r¯Ak,¯l ζ¯l∗⊗ζk + 4∂r2,r¯Ak,¯l ζl∗⊗ζ¯k
+ 2 Ak,p¯Rp,¯l − Ap,¯l Rp,k¯ ζ¯l∗⊗ζk
(cid:0) (cid:1)
+ 2 Ak,p¯Rl,p¯ − Ap,¯l Rk,p¯ ζl∗⊗ζ¯k .
(cid:0) (cid:1)
Expandingthesecondorderderivativeweinferforallindicesk,l the expression
∂r2,r¯Ak,¯l = 2∂r4,r¯,k¯,¯l fˆ + 2∂¯l Rp,k¯ ∂p¯fˆ + 2Rp,k¯ ∂¯l2,p¯fˆ.
By plunging this in the previous expression and canceling the adequate therms
we obtain
∆A = ∂¯∇∆fˆ + 2∂ Ric∗·∇fˆ
E
+ 4∂¯l2,p¯fˆ Rp,k¯ ζ¯l∗⊗ζk + 4Rk,p¯ ∂p2,l fˆ ζl∗⊗ζ¯k
+ 4∂k¯2,p¯fˆ Rp,¯l ζ¯l∗⊗ζk + 4Rl,p¯ ∂p2,k fˆ ζl∗⊗ζ¯k ,
8
and thus the identity
∆A=∂ ∇∆fˆ + 2∂ Ric∗·∇fˆ + Ric∗A + ARic∗ . (12)
E
In conclusion the identities (11), (12), (6) (at the level of scalar products) and
(8) imply
∆ |α∗|2 + |A|2 = 2|∇α∗|2 + 2|∇A|2
(cid:0) (cid:1)
− 2hα∗,∆Ric∗i + 2hA,∆Ai
= 2|∇α∗|2 + 2|∇A|2
∗
− 2 α∗, i∂∂ Scal + ∆fˆ
D h (cid:16) (cid:17)i E
+ 4 α∗,(RicRm)∗ − Ric∗ Ric∗
(cid:10) (cid:11)
+ 2 A,∂∇∆fˆ + 2∂ Ric∗·∇fˆ
E
D E
+ 4 A2,Ric∗ .
(cid:10) (cid:11)
2.3 Evolution of the squared norms
The previous expression of the Laplacian of the squared norms combined with
the formulas (7) and (9) provides the identity
(cid:3) |α∗|2 + |A|2 = 2|∇α∗|2 + 2|∇A|2
(cid:0) (cid:1)
∗
− 2 α∗, i∂∂ 2∆fˆ − |∇fˆ|2 + Scal
D h (cid:16) (cid:17)i E
+ 2 A,∂ ∇ 2∆fˆ − |∇fˆ|2 + Scal + 2fˆ
D (cid:16) (cid:17)E
+ 4 α∗,(Ric∗Rm)∗ − 4hα∗,Ric∗i .
(cid:10) (cid:11)
Thus if we set
2Hˆ := 2H(gˆ,fˆ) = (cid:3)fˆ + 2fˆ,
we infer the evolution formula
(cid:3) |α∗|2 + |A|2 = 2|∇α∗|2 + 2|∇A|2
(cid:0) (cid:1)
− 4 α∗,(i∂∂Hˆ)∗ + 4 A,∂∇Hˆ
D E D E
− 4hαRm,αi . (13)
9
2.4 Integration by parts along the K¨ahler-Ricci flow
This last step is the key part of the proof of the second variationof Perelman’s
W-functional along the K¨ahler-Ricci flow. Deriving the identity
0 = ∆Hˆ − ∇Hˆ ·∇fˆ e−fˆdV ,
gˆ
ZX(cid:16) (cid:17)
we obtain
0= (cid:3) ∆Hˆ − ∇Hˆ ·∇fˆ e−fˆdV , (14)
gˆ
ZX (cid:16) (cid:17)
thanks to (4). We expand now the integrand therm in (14). We observe first
the identity
(cid:3)∆Hˆ = ∆(cid:3)Hˆ + 2∆Hˆ − 2 Ric,i∂∂Hˆ
D E
= 2∆H − 2 Ric,i∂∂Hˆ ,
2
D E
where
2H := (cid:3)Hˆ + 2Hˆ .
2
(Formoredetailsonthistype ofcomputationsseethe proofoftheformula(19)
in the subsection 4.1 of the appendix). We compute next the heat therm
(cid:3)(∇Hˆ ·∇fˆ).
We observe first the expression of the time derivative
∂ ∂
∇Hˆ ·∇fˆ = dHˆ ·∇fˆ
∂t ∂t
(cid:16) (cid:17) (cid:16) (cid:17)
∂ ∂
= d Hˆ ·∇fˆ + dHˆ · ∇fˆ
(cid:18)∂t (cid:19) ∂t
∂ ∂
= ∇ Hˆ ·∇fˆ + ∇Hˆ ·∇ fˆ − fˆ
(cid:18)∂t (cid:19) (cid:18)∂t (cid:19)
+ Ric ∇fˆ,J ∇Hˆ .
0
(cid:16) (cid:17)
We expand the Laplacian
∆ ∇Hˆ ·∇fˆ = 8∂r2,r¯ ωˆk,¯l Hˆl fˆk¯ + fˆl Hˆk¯
(cid:16) (cid:17) h (cid:16) (cid:17)i
= −8∂r2,r¯ωˆk,¯l Hˆl fˆk¯ + fˆl Hˆk¯
(cid:16) (cid:17)
+ 8∂r2,r¯ Hˆl fˆk¯ + fˆl Hˆk¯
(cid:16) (cid:17)
10