Table Of ContentThe total second variation of Perelman’s
-functional
W
2
1
0
2
NEFTON PALI
n
a
J January 5, 2012
4
]
G Abstract
D We show avery simple and general total second variation formula for
. Perelman’s W-functional at arbitrary points in the space of Riemannian
h
metrics. Moreover we perform a study of the properties of thevariations
t
a of K¨ahler structures. We deduce a quite simple and general total sec-
m ond variation formula for Perelman’s W-functional with respect to such
variations. In this case the main therm in the formula depends strongly
[
on the variation of the complex structure. We discover also convexity
1 of Perelman’s W-functional along particular variations over points with
v non-negative Bakry-Emery-Riccitensor.
9
6
9 1 Introduction
0
.
1 TotalsecondvariationformulasforPerelman’sW-functionalatashrinkingRicci
0
solitonpointwereobtainedindependently byCao-Hamilton-Ilmanenen[C-H-I],
2
Cao-Zhu [Ca-Zhu] and Tian-Zhu [Ti-Zhu]. The work of Cao-Zhu [Ca-Zhu] is
1
: based on the previous work of Cao-Hamilton-Ilmanenen [C-H-I]. Important
v
applications to the stability and the convergence of the K¨ahler-Ricci flow over
i
X FanomanifoldsweregivenbyTian-Zhu[Ti-Zhu]andbyTian-Zhang-Zhang-Zhu
r [T-Z-Z-Z]. The second variation formulas obtained by the previous authors are
a
of different nature from ours. We explain now our set-up and results.
Let (X,g) be a compact oriented (for simplicity) Riemannian manifold of
dimension m and let f be a smooth real valued function over X. We remind
that Perelman’s W-functional is defined (up to a constant) by the formula
W(g,f) := |∇ f|2 + Scal + 2f − m e−fdV .
Z g g g g
X(cid:0) (cid:1)
LetalsoΩ>0 be a smoothvolumeformoverX. We remindthata remarkable
result due to Perelman [Per] asserts that the first variation of the functional
W (g) := W(g,log(dV /Ω)),
Ω g
1
at a point g is given by the formula
D W (v) = hv, g − Ric (Ω)i Ω,
g Ω Z g g
X
whereRic (Ω)istheΩ-Bakry-Emery-Riccitensor. (Intheappendix7.2wegive
g
a straightforward proof of this formula based on the variation formula for the
Ω-Bakry-Emery-Ricci tensor.) This formula shows that Perelman’s modified
Ricci flow type equation is the gradient flow of the functional W with respect
Ω
to the metric
G := h·,·i Ω
Z g
X
overthespaceofRiemannianmetrics. WeremindthatthespaceofRiemannian
metricsequippedwiththismetricisanon-positivelycurvedspacewithgeodesics
given by the formula
gt = g0etg0−1g˙0 .
Thus it is natural to compute the total second variation of the functional W
Ω
with respect to the metric G. In this paper we obtain the following quite
simple and general second variation result at arbitrary points in the space of
Riemannian metrics.
Theorem 1. Let (X,g) be a compact and orientable Riemannan manifold and
let Ω > 0 be a smooth volume form. Along any smooth curve (g ) of
t t∈(−ε,ε)
Riemannian metrics hold the second variation formula
∇ DW (g )(g˙ ,g˙ ) = g˙ ·Ric∗ (Ω), g˙ Ω
G Ω t t t ZX(cid:10) t gt t(cid:11)gt
+ 1 ∇ˆ g˙ 2 − ∇ g˙ 2 Ω, (1)
ZX(cid:20)6 (cid:12) gt t(cid:12)gt (cid:12) gt t(cid:12)gt(cid:21)
(cid:12) (cid:12) (cid:12) (cid:12)
where Ric∗ (Ω) denotes the endomorphism section associated to the Ω-Bakry-
gt
Emery-Ricci tensor and ∇ˆ denotes the symmetrization of ∇ acting on sym-
g g
metric 2-tensors.
This formula suggest naturally the introduction of the vector space
:= v ∈C∞ X,S2T∗ | ∇ v∗ = 0 ,
Fg n (cid:0) R X(cid:1) TX,g g o
where ∇ denotes the covariant exterior derivative acting on T -valued dif-
TX,g X
ferentialforms and v∗ :=g−1v denotes the endomorphismsection associatedto
g
v. Indeed we observe the following corollary.
Corollary 1. Let (X,g) be a compact and orientable Riemannan manifold and
let Ω > 0 be a smooth volume form. Then for all v ∈ hold the second
g
F
variation formula
∇ DW (g)(v,v)= v·Ric∗(Ω), v + 1 ∇ v 2 Ω. (2)
G Ω Z (cid:20) g g 2 g g(cid:21)
X (cid:10) (cid:11) (cid:12) (cid:12)
(cid:12) (cid:12)
2
In particular if g satisfies the inequality Ric (Ω)>εg, for some ε∈ , then
g >0
R
the bilinear form
∇ DW (g) : × −→ ,
G Ω g g
F F R
is positive definite and for all v ∈ hold the inequality
g
F
∇ DW (g)(v,v) > ε|v|2 + 1 ∇ v 2 Ω > 0. (3)
G Ω Z (cid:20) g 2 g g(cid:21)
X (cid:12) (cid:12)
(cid:12) (cid:12)
An other surprising consequence of the proof of Theorem 1 is a drastically
simple secondvariationformula for the functional W over the space of K¨ahler
Ω
metrics with respect to a fixed complex structure.
Lemma 1. Let (X,g,J) be a compact K¨ahler manifold and let Ω > 0 be a
smooth volume form. For all J-invariant v ∈ C∞ X,S2T∗ such that the
X
differential form vJ is d-closed hold the second variat(cid:0)ion foRrmu(cid:1)la
∇ DW (g)(v,v) = v·Ric∗(Ω), v Ω.
G Ω Z g g
X(cid:10) (cid:11)
InordertoobtainourgeneralsecondvariationformulaforthefunctionalW
Ω
withrespecttomoregeneralvariationsofK¨ahlerstructuresweneedtoperform
a detailed study of their properties. This is done in detail in the section 5. Our
main result is the following.
Theorem 2. (Main Result). Let (X,g,J) be a compact K¨ahler manifold, let
Ω > 0 be a smooth volume form and let (g ,J ) ,g = g ,J = J be a
t t t∈(−ε,ε) 0 0
smooth family of K¨ahler structures such that J˙ = (J˙)T. Then at the point g
t t gt
hold the second variation formula in the direction v :=g˙
0
∇ DW (g)(v,v) = Tr (v∗)2Ric∗(Ω) + 1 ∇0,1 (v∗)0,1 2 Ω
G Ω ZX(cid:26) R(cid:2) g g (cid:3) 2 (cid:12) g,J g J (cid:12)g(cid:27)
(cid:12) (cid:12)
1
− 4∂ (v∗)1,0 + ∂g (v∗)0,1, ∂g (v∗)0,1 Ω,
2ZXD TX,J g J TX,J g J TX,J g J Eg
where (v∗)1,0, (v∗)0,1 denote respectively the J-linear and J-anti-linear compo-
g J g J
nents of the endomorphism section v∗. In particular if v ∈ then hold the
g Fg
identity
∇ DW (g)(v,v) = Tr (v∗)2Ric∗(Ω) Ω
G Ω ZX R(cid:2) g g (cid:3)
+ 1 ∇ (v∗)0,1 2 + ∇0,1(v∗)0,1 2 Ω.
Z (cid:20)2 g g J g g,J g J g(cid:21)
X (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
WeobservethatthevariationsofK¨ahlerstructuressuchthatJ˙ =(J˙)T are
t t gt
quite natural in K¨ahler geometry (see [Don] for example). Moreover from the
3
pointofviewofapplicationsisseemstobeuselesstoconsidermoregeneraltype
of variations. It is easy to see that for this type of variations hold the identity
JJ˙ = (v∗)0,1. (See the section 5.) This shows that the main therm in the
0 g J
second variation expression depends strongly on the variation of the complex
structure. We pointout also that the -valuedvariations ofK¨ahler structures
g
F
enjoy the following remarkable property.
Proposition 1. Let (g ) be a smooth family of Riemannian metrics such
t t>0
that g˙ ∈ for all t>0 and let (J ) be a family of endomorphism sections
t Fgt t t>0
of T solution of the ODE
X
2J˙ = J g˙∗ − g˙∗J ,
t t t t t
with K¨ahler initial data (J ,g ). Then (J ,g ) is a smooth family of K¨ahler
0 0 t t t>0
structures.
Relevant applications of this fact will be presented in a forthcoming work.
We expectthat our mainresult willimply quite sharpstability statements. We
postpone this considerations in a forthcoming study.
2 The first variation of the Ω-Bakry-Emery-Ricci
tensor
Let Ω > 0 be a smooth volume form over an oriented Riemannian manifold
(X,g). We remind that the Ω-Bakry-Emery-Riccitensor of g is defined by the
formula
dV
g
Ric (Ω) := Ric(g) + ∇ d log .
g g
Ω
A Riemannian metric g is called a Ω-Shrinking Ricci soliton if g = Ric (Ω).
g
We define the the symmetrization of ∇ acting on symmetric tensors as the
g
operator
∇ˆ :C∞(SpT∗ )−→C∞ Sp+1T∗ ,
g M M
(cid:0) (cid:1)
p
∇ˆ α(ξ ,...,ξ ) = ∇ α(ξ ,ξ ,...,ξˆ,...,ξ ).
g 0 p g j 0 j p
Xj=0
We observe in fact that
∇ :C∞(SpT∗ )−→C∞(T∗ ⊗ SpT∗ ) .
g M M M
R
With this notations hold the following variation result.
Lemma 2. Let (g ) be a smooth family of Riemannian metrics over an ori-
t t
entable manifold oriented by a smooth volume form Ω >0. Then hold the first
variation formula
d
2 Ric (Ω) = eftdiv e−ftD g˙ ,
dt gt gt gt t
(cid:0) (cid:1)
where f := logdVgt and D := ∇ˆ − 2∇ .
t Ω g g g
4
Proof. A very large part of the proof is taken from a standard computation in
[Bes]. We include it here for readers convenience. We remind (see [Bes]) that
the first variation ∇˙ of the connection is given by the formula
gt
2g ∇˙ (ξ,η),µ =∇ g˙ (ξ,η,µ) + ∇ g˙ (η,ξ,µ) − ∇ g˙ (µ,ξ,η). (4)
t gt gt t gt t gt t
(cid:16) (cid:17)
This rewrites as
2g ∇˙ (ξ,η),µ = ∇ˆ g˙ (ξ,η,µ) − 2∇ g˙ (µ,ξ,η)
t gt gt t gt t
(cid:16) (cid:17)
= ∇ˆ g˙ (µ,ξ,η) − 2∇ g˙ (µ,ξ,η),
gt t gt t
thanks to the symmetry properties of ∇ˆ g˙ . We infer the identity
gt t
2g ∇˙ (ξ,η),µ = D g˙ (µ,ξ,η). (5)
t gt gt t
(cid:16) (cid:17)
Taking a covariantderivative of this identity we obtain
2g (∇ ∇˙ (ζ,ξ,η),µ)=∇ D g˙ (ζ,µ,ξ,η). (6)
t gt gt gt gt t
We observe now
n
d
Ric(g ) (ξ,η) = Tr R˙ (·,ξ)η = g R˙ (e ,ξ)η,e ,
(cid:18)dt t (cid:19) Rh gt i Xj=1 (cid:16) gt j j(cid:17)
for any g -orthonormal frame (e ) . This combined with the variation identity
t j j
R˙ (ξ,η)µ = ∇ ∇˙ (ξ,η,µ) − ∇ ∇˙ (η,ξ,µ), (7)
gt gt gt gt gt
(see [Bes]) implies the expression
2g R˙ (e ,ξ)η,e = 2g ∇ ∇˙ (e ,ξ,η),e
t gt j j t gt gt j j
(cid:16) (cid:17) (cid:16) (cid:17)
− 2g ∇ ∇˙ (ξ,e ,η),e
t gt gt j j
(cid:16) (cid:17)
= ∇ D g˙ (e ,e ,ξ,η) − ∇ D g˙ (ξ,e ,e ,η),
gt gt t j j gt gt t j j
thanks to the identity (6). Let (x ,...,x ) be g -geodesic coordinates centered
1 n t
at an arbitrary point p and set e := ∂ . Then the local frame (e ) is g (p)-
k ∂xk k k t
orthonormalatthepointpandsatisfies∇ e (p)=0forallj. Wetakenowthe
gt j
vector fields ξ and η with constant coefficients with respect to the g -geodesic
t
coordinates(x ,...,x ). Therefore∇ ξ(p)=∇ η(p)=0. Weinfertheidentity
1 n gt gt
at the space time point (p,t)
∇ D g˙ (ξ,e ,e ,η) = ∇ D g˙ (e ,g−1e∗,η)
gt gt t j j gt,ξ gt t j t j
(cid:2) (cid:3)
= ∇ [(Tr D g˙ )(η)].
gt,ξ gt gt t
5
We observe indeed the trivial identities
Tr g˙ = Tr g−1g˙
gt t t t
R(cid:0) (cid:1)
n
= e∗ g−1g˙ e
j t t j
Xj=1 (cid:0) (cid:1)
n
= g g−1e∗, g−1g˙ e
t t j t t j
Xj=1 (cid:0) (cid:0) (cid:1) (cid:1)
n
= g˙ (e ,g−1e∗).
t j t j
Xj=1
Derivingthis lastidentity wegetthe formulaη.Tr g˙ =Tr ∇ g˙ . Moreover
gt t g gt,η t
the identity (5) combined with (4) gives
D g˙ (e ,e ,η) = ∇ g˙ (e ,e ,η) + ∇ g˙ (η,e ,e ) − ∇ g˙ (e ,e ,η)
gt t j j gt t j j gt t j j gt t j j
= ∇ g˙ (η,e ,e ).
gt t j j
We deduce the identity (Tr D g˙ )(η) = Tr ∇ g˙ and thus
gt gt t g gt,η t
Tr D g˙ = dTr g˙ .
gt gt t gt t
We obtain the variation formula
d
2 Ric(g ) = div D g˙ − ∇ dTr g˙ . (8)
dt t gt gt t gt gt t
We observe now the identity
d
2 ∇ df = 2∇˙ df + 2∇ df˙ .
dt gt t gt t gt t
Time deriving the definition of the dual connection we obtain the identity
(∇˙ α)·η = − α·∇˙ η ,
gt,ξ gt,ξ
for any 1-form α. By applying this identity to α=df we infer
t
2∇˙ df (ξ,η) = − 2df ·∇˙ (ξ,η)
gt t t gt
= − 2g ∇˙ (ξ,η),∇ f
t gt gt t
(cid:16) (cid:17)
= −D g˙ (∇ f ,ξ,η),
gt t gt t
i.e. 2∇˙ df = −∇ f ¬D g˙ , where ¬ denotes the contraction operation.
gt t gt t gt t
Thus hold the formula
d
2 ∇ df = − ∇ f ¬D g˙ + ∇ dTr g˙ ,
dt gt t gt t gt t gt gt t
6
since 2f˙ = Tr g˙ . Using the identity (8) we obtain
t gt t
d
2 Ric (Ω) = div D g˙ − ∇ f ¬D g˙
dt gt gt gt t gt t gt t
= Tr (∇ D g˙ − df ⊗D g˙ )
gt gt gt t t gt t
= eftTr e−ft∇ D g˙ + de−ft ⊗D g˙
gt gt gt t gt t
(cid:0) (cid:1)
= eftTr ∇ e−ftD g˙ .
gt gt gt t
(cid:0) (cid:1)
This concludes the proof of the lemma.
We denote by P∗ the formal adjoint of an operator P with respect to a
g
metric g. We observe that the operator
P∗Ω := efP∗ e−f• ,
g g
(cid:0) (cid:1)
is the formal adjoint of P with respect to the scalar product h·,·i Ω. With
X g
this notations hold the following corollary R
Corollary 2. Let X be an orientable manifold oriented by a smooth volume
form Ω>0 and let (g ) be a smooth family of Riemannian metrics. Then hold
t t
the first variation formula
d Ric (Ω) = ∇∗Ω∇ g˙ − 1∇ˆ∗Ω∇ˆ g˙ .
dt gt gt gt t 6 gt gt t
Proof. We remind that ∇∗ = −div and that ∇ˆ∗ = 3∇∗ in restriction to
g g g g
symmetric 3-forms. (See the subsection7.1 inthe appendix). This lastidentity
implies ∇ˆ∗Ω =3∇∗Ω in restriction to symmetric 3-forms.
gt gt
3 Geodesics in the space of Riemannian metrics
The content of this section is well known. The presentation is adapted to our
particular situation. Moreover a formula below will be needed in the next sec-
tion.
The differential point of view. Let H := L2(X,S2T∗). We equip the
X
setM⊂H of smoothRiemannian metrics overX with thRe Riemannianmetric
G (u,v)= hu,vi Ω, (9)
g Z g
X
for all g ∈ M and u,v ∈ H. We observe now that any element u ∈ T∗ ⊗T∗
X X
can be considered as a morphism u : T → T∗. Thus we can define the
X X
endomorphism u∗ :=g−1u. With this notation hold the identity
g
hu,vi = Tr (u∗ v∗), (10)
g g g
R
7
for all u,v ∈ S2T∗. Using the identity (10) we compute now the Fr´echet
X
derivative R
D G:H×H→H∗,
g
of the metric G at a point g ∈ M. For this purpose let (g ) ⊂ M be a
s s∈(−ε,ε)
smooth curve and for notation simplicity let denote u∗ :=g−1u. Then hold the
s s
equalities
d d
hu,vi = Tr (u∗ v∗)
ds gs ds R s s
d d
= Tr u∗ v∗ + u∗ v∗ ,
(cid:18)ds s s s ds s(cid:19)
R
and d u∗ =−g˙∗u∗ since d g−1 =−g−1g˙ g−1. Thus
ds s s s ds s s s s
d
hu,vi = − Tr (g˙∗u∗v∗ + u∗g˙∗ v∗)
ds gs R s s s s s s
= − 2 Tr (g˙∗u∗ v∗),
s s s
R
since g˙ is also symmetric. Indeed we observe the elementary identities
s
Tr [(u∗g˙∗)v∗] = Tr [v∗(u∗g˙∗)]
s s s s s s
R R
= Tr [v∗(u∗g˙∗)]T
s s s s
R
= Tr (u∗g˙∗)Tv∗
s s s s
R(cid:2) (cid:3)
= Tr (g˙∗ u∗v∗),
s s s
R
where AT denotes the transpose of A with respect to g . So if we set g := g
s s 0
and h:=g˙ we infer the identity
0
D G(h,u)v = − 2 Tr (h∗ u∗v∗)Ω
g ZX R g g g
= − 2 hu∗ ,v Ω.
Z g g
X(cid:10) (cid:11)
Clearly the domain of D G is of type E×H, with E ⊂H a linear space dense
g
inside H. We remind now that the Levi-Civita connection ∇ = D+Γ of G
G G
is given by the formula
2Γ (g)(u,v) := G−1 D G(u,v) + D G(v,u) − D G(•,u)v .
G g g g g
h i
8
We explicit the therm
D G(u,v) + D G(v,u) − D G(•,u)v
g g g
= − 2 Tr u∗v∗ •∗ + v∗ u∗ •∗ − •∗ u∗ v∗ Ω
ZX R(cid:0) g g g g g g g g g(cid:1)
= − 2 Tr (u∗ v∗ •∗)Ω
ZX R g g g
= − Tr u∗ v∗ •∗ + v∗ u∗ •∗ Ω
ZX R(cid:0) g g g g g g(cid:1)
= − uv∗ + vu∗,• Ω.
Z g g g
X(cid:10) (cid:11)
We infer the identity
2Γ (g)(u,v) = − uv∗ − vu∗ . (11)
G g g
We deduce that the equation of the geodesics g¨ +Γ (g )(g˙ ,g˙ )=0, writes as
t G t t t
g¨ − g˙ g−1g˙ = 0,
t t t t
(we consider g as a morphism g : T → T∗) or as g¨∗−(g˙∗)2 = 0. Moreover
t t X X t t
time deriving the identity g˙ = g g˙∗, which defines the endomorphism g˙∗,we
t t t t
obtain the expression
d
g¨ = g˙ g˙∗ + g g˙∗,
t t t t dt t
and thus
d
0 = g¨∗ − (g˙∗)2 = g˙∗. (12)
t t dt t
But the equation (12) implies the identity g−1g˙ = g−1g˙ . Thus the geodesic
t t 0 0
writes explicitly as
gt = g0etg0−1g˙0 . (13)
We show in the subsection 7.3 of the appendix that the Riemannian space
(M,G) is non-positively curved.
The metric point of view. For any g ,g ∈ M we consider the set of
0 1
curves
C1(g ,g ) := h∈C1([0,1],M)| h = g , h = g ,
0 1 0 0 1 1
(cid:8) (cid:9)
and the distance function
1 1/2
d (g ,g ) := inf dt |h˙ |2 Ω .
G 0 1 h∈C1(g0,g1)Z0 (cid:20)ZX t ht (cid:21)
9
Let (g ) ⊂M be the unique geodesic joining g to g . Then
t t∈ 0 1
R
1/2
d (g ,g ) = |g˙ |2 Ω
G 0 1 (cid:20)Z 0 g0 (cid:21)
X
1/2
= Tr (g−1g˙ )2Ω
(cid:20)ZX R 0 0 (cid:21)
1/2
= Tr log(g−1g ) 2Ω ,
(cid:20)ZX R(cid:2) 0 1 (cid:3) (cid:21)
thanksto the expressionofthe geodesics(13). We observenow thata sequence
{g } ⊂Misd -Cauchyifandonlyifthesequence{log(g−1g )} isL2-Cauchy.
k k G 0 k k
Indeed hold the identity
1/2
d (g ,g ) = Tr log(g−1g ) − log(g−1g ) 2Ω .
G k k+l (cid:20)ZX R(cid:2) 0 k+l 0 k (cid:3) (cid:21)
This is because for any g -symmetric endomorphisms A,B,C, with A,B > 0,
0
the identity
Tr (ABC) = Tr (BAC),
R R
implies
Tr [log(AB)C] = Tr [(logA + logB)C] .
R R
We obtain
1/2
d (g ,g ) = log(g−1g ) − log(g−1g ) 2 Ω ,
G k k+l (cid:20)Z 0 k+l 0 k g0 (cid:21)
X(cid:12) (cid:12)
(cid:12) (cid:12)
since the endomorphism
log(g−1g ) − log(g−1g ),
0 k+l 0 k
is g0-symmetric. We infer that the metric completion MdG of (M,dG) is given
by
MdG ≡ g ∈Mes(X,S2T∗) | g > 0, log(g−1g)∈L2(X,End(T )) ,
X /a.e 0 X
n R o
where the notation /a.e means the almost everywhere equivalence relation.
Moreover is clear that
MdG = MdG .
Using quite elementary relaxation considerations we can show that the metric
space(MdG,dG)is a nonpositively curvedlengthspacein the sense ofAlexan-
drov.
10
