Vol. 22, No. 2 (2016), pp. 157–176.
EIGENVALUES VARIATION OF THE P-LAPLACIAN
UNDER THE RICCI FLOW ON SM
Shahroud Azami
1and Asadollah Razavi
21Department of Mathematics, Faculty of Science, Imam khomeini
international university, Qazvin, Iran azami@sci.ikiu.ac.ir
2Department of Mathematics and Computer Science, Amirkabir university
of technology,Tehran,Iran
arazavi@aut.ac.ir
Abstract.Let (M, F) be a compact Finsler manifold. Studying the eigenvalues and eigenfunctions for the linear and nonlinear geometric operators is a known problem. In this paper we will consider the eigenvalue problem for thep-laplace operator for Sasakian metric acting on the space of functions onSM. We find the first variation formula for the eigenvalues ofp-Laplacian onSMevolving by the Ricci flow onM and give some examples.
Key words and Phrases: Ricci flow, Finsler manifold,p-Laplace operator
Abstrak.Misalkan (M, F) adalah suatu manifold Finsler kompak. Sejauh ini telah dipelajari fungsi eigen dan nilai eigen untuk operator-operator geometri linier dan non linier. Dalam paper ini kami akan memperhatikan masalah nilai eigen untuk operatorp-Laplace untuk metrik Sasakian yang berlaku pada ruang fungsi diSM. Kami memperoleh rumus variasi pertama dan memberikan beberapa contoh untuk nilai eigen darip-Laplacian padaSMyang melibatkan aliran Ricci padaM.
Kata kunci: aliran Ricci, manifold Finsler, operatorp-Laplace
1. Introduction
For a compact Finsler manifold (M, F), studying the eigenvalues of geometric operators plays a powerful role in geometric analysis. In the classical theory of the Laplace or p-Laplace equation several main parts of mathematics are joined in a fruitful way: Calculus of Variation, Partial Differential Equation, Potential Theory,
2000 Mathematics Subject Classification: Primary 58C40; Secondary 53C44. Received: 22-09-2015, revised: 14-11-2016, accepted: 14-11-2016.
158
Analytic Function. Recently, there are many mathematicians who have investigated properties of the eigenvalues ofp-Laplacian on Finsler manifolds and Riemannian manifolds to estimate the spectrum in terms of the other geometric quantities of the manifold. (see [3, 4, 9, 11, 18, 20]).
Also, geometric flows have been a topic of active research interest in math-ematics and other sciences (see [5, 8, 10, 12, 13, 15, 16]). Hamilton’s Ricci flow ( [6]) is the best known example of a geometric evolution equation. The Ricci flow is related to dynamical systems in the infinite-dimensional space of all metrics on a given manifold. One of the aims of such flows is to obtain metrics with special properties. Special cases arise when the metric is invariant under a group of trans-formations and this property is preserved by the flow.
Let M be a manifold with a Finsler metricg0 (or F0), the family g(t) (or Ft) of
Finsler metrics on M is called an un-normalized Ricci flow when it satisfies the equations
∂logF
∂t =−Ric, (1)
with the initial condition
F(0) =F0
or equivalently satisfies the equations
∂gij
∂t =−2Ricij, g(0) =g0 (2) whereRicis the Ricci tensor of g(t),Ricij = (21F2Ric)yiyj. In fact Ricci flow is a system of partial differential equations of parabolic type which was introduced by Hamilton on Riemannian manifolds for the first time in 1982 and Bao (see [2, 17]) studied Ricci flow equation in Finsler manifold. The Ricci flow has been proved to be a very useful tool to improve metrics in Finsler geometry, whenM is compact. One often considers the normalized Ricci flow
∂logF
∂t =−Ric+ 1 vol(SM)
Z
SM
Ric dv, F(0) =F0. (3)
or
∂gij
∂t =−2Ricij+ 2 vol(SM)
Z
SM
Ric dvgij, g(0) =g0 (4)
2. Preliminaries
Let M be an n-dimensional C∞ manifold. For a point
x ∈ M, denote by TxM the tangent space atx∈M,and byT M =∪x∈MTxM the tangent bundle of
M. Any element ofT M has the form (x, y), wherex∈M andy∈TxM.
Definition 2.1. A Finsler metric on a manifoldM is a functionF :T M0→[0,∞)
which has the following properties:
(i): F(x, αy) =αF(x, y), ∀α >0; (ii): F(x, y)isC∞ on
T M0 ;
(iii): For any non-zero tangent vectory∈TxM, the associated quadratic form
gy :TxM ×TxM →Ron T M is an inner product, where
gy(u, v) :=
1 2
∂2 ∂s∂r
F2(x, y+su+rv)
s=r=0 .
The pair(M, F)is called a Finsler manifold.
Let us denote bySxM the set consisting of all rays [y] :={λy|λ >0}, where
y∈TxM0. The Sphere bundle ofM, i.e. SM, is the union of SxM’s :
SM=∪
xSxM
SMhas a natural (2n−1)-dimensional manifold structure. We denote the elements ofSM by (x,[y]) wherey∈TxM0. If there is not any confusion we write (x, y) for
(x,[y]). In a local coordinate system xi, yiwe haveg
ij(x, y) =12
∂2F2
∂yi∂yj(x, y) and (gij) := (g
ij)−1. The geodesics ofF are characterized locally by
d2xi
dt2 + 2G
i(x,dx
dt) = 0 where
Gi= 1 4g
il
2∂gjl
∂xk −
∂gjk
∂xl
yjyk. (5)
Definition 2.2. The coefficients of the Riemann curvature Ry=Rikdxi⊗ ∂ ∂xi are
given by
Rik:= 2∂G
i
∂xk −
∂2Gi ∂xj∂yky
j+ 2Gj ∂2Gi
∂yj∂yk −
∂Gi ∂yj
∂Gj
∂yk (6)
andRijk:=1 3
∂Ri j
∂yk −
∂Ri k
∂yj
,Rj kli :=1 3
∂2Ri k
∂yj∂yl−
∂2Ri l
∂yj∂yk
.
The Ricci scalar function ofF is given byRic:= 1
F2Rii. A companion of the
Ricci scalar is the Ricci tensor
Ricij :=
1
2F 2Ric
yiyj
160
Definition 2.3. A Finsler metric is said to be an Einstain metric if the Ricci scalar function is a function of xalone,equivalently Ricij =R(x)gij (see [14, 19]).
Definition 2.4. Let (M, F)be a Finsler manifold, the Sasakian metriceg of g on
T M0 is defined as
Remark. The Levi-Civita connection ∇e on T M0 with respect to the Sasakian metricegis locally expressed as follows:
e define the divergencedivX˜ as follows:
Definition 2.6. According to the above definition, the gradient of a functionf is
˜
∇f =gijδf δxi
δ δxj +F
2 gij∂f
∂yi
∂
∂yj (11)
therefore, the norm of ˜∇f with respect to the Riemannian metriceg is given by
|∇˜f|2=eg(∇ef,∇ef) =gij δf δxi
δf δxj +F
2 gij∂f
∂yi
∂f
∂yj. (12)
Definition 2.7. Let M be a compact Finsler manifold. The Laplace operator of f
on T M is defined as follows:
∆f = gij( ∂ 2f ∂xi∂xj −
∂Grj ∂xi
∂f ∂yr −G
r j
∂2f ∂xi∂yr −G
s i
∂2f ∂ys∂xj +G
r jGsi
∂2f ∂ys∂yr)
+ gijF2 ∂ 2f ∂yi∂yj +g
ij(Ck i j+
1 2R
k ij)
δf δxk −F g
ijFk i j
δf
δxk (13)
− F gijCi jk ∂f ∂yk −F
2gijg
ih(Gj lh −Fj lh)glk
δf δyk.
Definition 2.8. Let M be a compact Finsler manifold. Thep-Laplace operator of
f :SM→R,f ∈W1,p(SM) for1< p <∞ is defined as follows:
△pf = div(|∇ef|p
−2∇ef) (14)
= |∇ef|p−2∆f+ (p−2)|∇ef|p−4(Hessf)(∇ef,∇ef)
where
(Hessf)(X, Y) =∇e(∇ef)(X, Y) =Y.(X.f)−(∇eYX).f, X, Y ∈ X(SM)
and in local coordinate, we have:
(Hessf)(∂i, ∂j) =∂i∂jf −Γekij∂kf.
Note. Iff is a function of xalone, or suppose that is the lifting of f : M → R then
∆f =gij
∂2f ∂xi∂xj −Γe
k ij
∂f ∂xk
(15)
whereΓek
ij is christoffel symbol of ∇e.
2.1. Eigenvalues of the p-Laplacian.
Definition 2.9. Let(Mn, F)be a compact Finsler manifold andf :SM →R. We
say thatλis an eigenvalue of thep-Laplace operator whenever
e
g∆
pf+λ|f|p−2f = 0 (16)
then f is said to be the eigenfunction associated to λ, or equivalently they satisfy in
Z
SM
|∇ef|p−2<∇ef,∇eϕ > dv=λZ
SM
|f|p−2f ϕdv ∀ϕ∈W1,2
162
whereW01,p(SM)is closure of C∞
0 (SM)in SobolevW1,p(SM).
By substitutionϕ=f in (17) we have:
λ= R
SM|∇ef| pdv
R
SM|f|pdv
. (18)
Normalized eigenfunctions are defined as follows: Z
SM
f|f|p−2dv= 0,Z
SM
|f|pdv= 1. (19)
Suppose that (Mn, F
t) is a solution of the Ricci flow on the smooth manifold
(Mn, F0) in the interval [0, T) and
λ(t) = Z
SM
|∇ef(x, y)|pdv
t (20)
defines the evolution of an eigenvalue of P-Laplacian under the variation of Ft
whose eigenfunction associated to λ(t) is normalized.Suppose that for any metric g(t) onMn
Specp(eg) ={0 =λ0(g)≤λ1(g)≤λ2(g)≤...≤λk(g)≤...}
is the spectrum of ∆p =eg∆p. In what follows we assume the existence and C1
-differentiability of the elementsλ(t) and f(t), under a Ricci flow deformationg(t) of a given initial metric. We prove some propositions about the problem of the spectrum variation under a deformation of the metric given by a Ricci flow equation.
3. Variation of λ(t)
In this part, we will give some useful evolution formulas forλ(t) under the Ricci flow. Let (Mn, F
t),t∈[0, T),be a deformation of Finsler metricF0. Assume
that λ(t) is the eigenvalue of ∆p,f =f(x, y, t) satisfies
∆pf+λ|f|p−2f = 0
andRSM|f|pdv= 1, using (12), we have:
d dt|∇ef|
2 = ∂
∂t(g
ij)δf
δxi
δf δxj +g
ij ∂
∂t( δf δxi)
δf δxj
+ gijδf δxi
∂ ∂t(
δf δxj) +
∂(F2) ∂t g
ij∂f
∂yi
∂f
∂yj (21)
+ F2∂ ∂t(g
ij)∂f
∂yi
∂f ∂yj + 2F
2gij∂f ′
∂yi
∂f ∂yj.
where
∂ ∂t(g
ij) =−gilgjk ∂
and
therefore, a substitution of (22) and (23) in (21), implies that:
Proposition 3.1. Let (Mn, F
On the other hand we have
d
Now, we get the following two integrability conditions:
164
From the un-normalized Ricci flow, we can then write
Using (26) we obtain
p
We have thus proved the following proposition:
Proposition 3.2. Let (Mn, F
t)) be a solution of the un-normalized Ricci flow on the
smooth Finsler manifold (Mn, F0). Ifλ(t) denotes the evolution of an eigenvalue
under the Ricci flow, then
dλ
166
Note. Letf :SM→Rbe a lifting off :M →R. We have: dλ
dt = p Z
SM
Ric(∇ef,∇ef)|∇ef|p−2 dv
+ Z
SM
(λ|f|p− |∇ef|p)−2gijRic
ij+nRic dv
and in this case, if−2gijRicij+nRicis a constant, then
dλ dt =p
Z
SM
Ric(∇ef,∇ef)|∇ef|p−2 dv.
Corollary 3.3. Let (Mn, F
t)be a solution of the un-normalized Ricci flow on the
smooth Riemannian manifold(Mn, F0), i.e. F
t,F0are Riemannian metric. Ifλ(t)
denotes the evolution of an eigenvalue under Ricci flow, then:
dλ
dt = p Z
SM
Ric(∇ef,∇ef)|∇ef|p−2 dv+
Z
SM
R(λ|f|p− |∇ef|p)dv
−p Z
SM
gij ∂ ∂t(G
i r)
∂f ∂yr
δf δxj|∇ef|
p−2dv (31)
−p
n Z
SM
RF2gij∂f ∂yi
∂f ∂yj|∇ef|
p−2dv
whereR is the scalar curvature of M.
Proof:
IfFtis the Riemannian metric, then
Ric= 1
nR, (32)
and
2gijRicij−nRic=R, (33)
therefore (31) is obtained by replacing (32) and (33) in (29). ✷
Corollary 3.4. Let (M2, F
t)be a solution of the un-normalized Ricci flow on the
smooth Riemannian surface(M2, F0). Ifλ(t)denotes the evolution of an eigenvalue
under the Ricci flow, then:
dλ dt =
p 2 Z
SM
R|∇ef|pdv+
Z
SM
R(λ|f|p− |∇ef|p)dv
−p Z
SM
gij ∂ ∂t(G
i r)
∂f ∂yr
δf δxj|∇ef|
p−2dv
−p
n Z
SM
RF2gij ∂f ∂yi
∂f ∂yj|∇ef|
p−2dv
whereR is the scalar curvature of M.
Proof: In dimensionn= 2, for a Riemannian manifold, we have:
Ric= 1
hence the corollary is obtained by replacing (34) in (29). ✷
Corollary 3.5. Let (Mn, F
t)be a solution of the un-normalized Ricci flow on the
smooth homogenous Riemannian manifold (Mn, F0). Ifλ(t) denotes the evolution
of an eigenvalue under the Ricci flow, then:
dλ
dt = p Z
SM
Ric(∇ef,∇ef)|∇ef|p−2dv
−p Z
SM
gij ∂ ∂t(G
i r)
∂f ∂yr
δf δxj|∇ef|
p−2dv
−Rp n
Z
SM
F2gij∂f ∂yi
∂f ∂yj|∇ef|
p−2dv
whereR is the scalar curvature of M.
Proof:
Since the evolving metric remains homogenous and a Riemannian homoge-nous manifold has constant scalar curvature, so the corollary is obtained by (29).✷ Now, we give a variation of λ(t) under the normalized Ricci flow which is similar to the pervious proposition.
Proposition 3.6. Let (Mn, F
t) be a solution of the normalized Ricci flow on the
smooth Finsler manifold (Mn, F0). Ifλ(t) denotes the evolution of an eigenvalue
under Ricci flow, then:
dλ
dt = (n−p)rλ+p Z
SM
Ric(∇ef,∇ef)|∇ef|p−2 dv
+ Z
SM
(λ|f|p− |∇ef|p)2gijRic
ij−nRic dv (35)
−p Z
SM
gij ∂ ∂t(G
s i)
∂f ∂ys
δf δxj|∇ef|
p−2 dv
−p Z
SM
F2(Ric−r)gij∂f ∂yi
∂f ∂yj|∇ef|
p−2 dv
where f is the associated normalized evolving eigenfunction,r=
R
SMRic dv
vol(SM) .
Proof: In the normalized case, the integrability conditions read as follows
p Z
SM
f′f|f|p−2 dv=
Z
SM
|f|p2gijRic
ij−nr−nRic dv. (36)
Since
d dt(dv) =
168
using (24), (27) and the above equation, we can then write
dλ
i) is obtained by replacing F ′
and g′
ij from (3) and (4), respectively, in
(27). Thus the proposition is obtained by replacing (39) in (38).✷ Similar to un-normalized case we have the following corollaries:
Corollary 3.7. Let (Mn, F
t) be a solution of the normalized Ricci flow on the
smooth Riemannian manifold (Mn, F0). If λ(t)denotes the evolution of an eigen-value under Ricci flow, then:
dλ
whereR is the scalar curvature of M.
Corollary 3.8. Let (M2, Ft) be a solution of the normalized Ricci flow on the
under Ricci flow, then:
dλ
dt = (2−p)rλ+ p 2 Z
SM
R|∇ef|pdv
+ Z
SM
R(λ|f|p− |∇ef|p)dv−p Z
SM
gij ∂ ∂t(G
s i)
∂f ∂ys
δf δxj|∇ef|
p−2dv
−p Z
SM
(R 2 −r)F
2gij ∂f
∂yi
∂f ∂yj|∇ef|
p−2dv
whereR is the scalar curvature of M. Corollary 3.9. Let (Mn, F
t) be a solution of the normalized Ricci flow on the
smooth homogenous Riemannian manifold (Mn, F0). Ifλ(t) denotes the evolution
of an eigenvalue under Ricci flow, then:
dλ
dt = (n−p)rλ+p Z
SM
Ric(∇ef,∇ef)|∇ef|p−2dv
−p Z
SM
gij ∂ ∂t(G
s i)
∂f ∂ys
δf δxj|∇ef|
p−2 dv
−p Z
SM
(1
nR−r)F 2
gij∂f ∂yi
∂f ∂yj|∇ef|
p−2 dv
whereR is the scalar curvature of M.
4. Examples
In this section, we will find the variational formula for some of Finsler mani-folds.
Example 4.1. Let (Mn, F0) be an Einstein manifold i.e. there exists a constant
a such that Ric(F0) = aF2
0. Therefore Ricij(g0) = agij(0). Assume we have a
solution to the Ricci flow which is of the form
g(t) =u(t)g0, u(0) = 1
whereu(t)is a positive function. Now (2) implies that
u(t) =−2at+ 1,
so that we have
g(t) = (1−2at)g0
which says that g(t)is an Einstein metric. On the other hand it is easily seen that
Ric(g(t)) = Ric(g0) =ag0= a 1−2atg(t),
Ric(gt) =
1
1−2atRic(g0) = a 1−2at, Ft2 = (1−2at)F02
therefore
2gijRicij−nRic=
170
and
Ric(∇ef,∇ef) = a
1−2ateg(∇ef,∇ef) = a
1−2at|∇ef| 2.
Also
Gi(t) = 1 4g
il{2∂gjl
∂xk −
∂gjk
∂xl }y jyk
= 1 4(g0)
il{2∂(g0)jl
∂xk −
∂(g0)jk
∂xl }y
jyk =Gi(0)
therefore
∂ ∂t(G
i r) = 0.
Using the un-normalized Ricci flow equation (2) and (29) ,we obtain the following relation:
dλ
dt = p Z
M
a
1−2at|∇ef|
pdv+λ
Z
M
|f|p an
1−2atdv
−
Z
M
an 1−2at|∇ef|
pdv−p
Z
SM
a 1−2atF
2 gij ∂f
∂yi
∂f ∂yj|∇ef|
p−2 dv
= pa
1−2at
λ−
Z
SM
F2gij ∂f ∂yi
∂f ∂yj|∇ef|
p−2dv
Now, If we suppose that gt=u(t)g0,u(0) = 1is a solution of the normalized Ricci
flow and r= V ol(1SM)RSMRicdv, then from (3) we have
u′g0=−2ag0+ 2rug0.
It implies that
u(t) = a r(1−e
2rt) +e2rt.
So that we have:
g(t) =na r(1−e
2rt) +e2rto
and
Ric(g(t)) =a a r(1−e
2rt) +e2rt
−1 g0
Ric(g(t)) =a
a r(1−e
2rt) +e2rt
−1
2gijRicij− Ricij =an
a r(1−e
2rt) +e2rt
−1 ,
also
Gi(t) =Gi(0),
therefore
∂Gr i
Using (35), we obtain the following:
dλ
dt = (n−p)rλ+paλ
a r(1−e
2rt) +e2rt
−1
− p Z
SM
a
a r(1−e
2rt) +e2rt
−1
−r
(yi∂f ∂yi
)2|∇ef|p−2 dv
=
(n−p)r+pa
a r(1−e
2rt) +e2rt
−1
λ
− p
a
a r(1−e
2rt) +e2rt
−1
−r Z
SM
F2gij ∂f ∂yi
∂f ∂yj|∇ef|
p−2dv.
Remark. Let (Mn, F0) be a Finsler manifold of dimensionn≥3. Suppose that the
flag curvaturek=k(x) is isotropic and a function ofx∈M alone thenk=constant and therefore (Mn, F0) is Einstein and the variation of its eigenvalues is similar to example (4.1).
Definition 4.2. A Finsler metric on an n-dimensional manifold is called a weak Einstein metric if
Ric= (n−1){3η
F +σ}F 2
whereη is a1-form and σ=σ(x)is scalar function.
Example 4.3. If we suppose that Ft=u(t)F0,u(0) = 1is a solution of the Ricci
flow, then:
Ric(Ft) =
Ric(Ft)
F2
t
= Ric(F0) (u(t))2F2
0
=(n−1){ 3η0
F0 +σ0}F
2 0 (u(t))2F2
0
= (n−1){ 3η0
F0 +σ0}
(u(t))2
Now the Ricci flow (1) implies that
−(n−1){
3η0 F0 +σ0}
(u(t))2 =−Ric= ∂logF
∂t = F′
t
Ft
= u
′(
t) u(t)
or equivalently
uu′
= (n−1){3η0
F0 +σ0}
By integration we have:
u2(t) =−2(n−1){3η0
F0 +σ0}t+c
with condition u(0) = 1 we have:
u2(t) = 1−2(n−1){3η0
F0 +σ0}t
therefore
Ft2=
1−2(n−1){3η0
F0 +σ0}t
172
and
Ric(∇ef,∇ef) = 1
2(n−1)g
ilgjs(3η0F0)
ylys+ 2σ0(g0)ls e∇if∇ejf. (42)
Also we have
2gijRicij−nRic= (n−1)gij(3η0F0)yiyj+ 2σ0(g0)ij −n
(n−1){3η0 F0 +σ0}
1−2(n−1){3η0 F0 +σ0}t
.
(43)
Note that
gij(t) = (1−2(n−1)σ0t)(g0)ij−6(n−1)t
∂η ∂yi
∂F
∂yj. (44)
By replacing (42),(43) and (44) in (29) we obtain the variation of an eigenvalue. Now, if we suppose thatFt=u(t)F0,u(0) = 1is a solution of the normalized Ricci
flow and r= V ol(1SM)RSMRic dv, then from (3), we have:
r−(n−1)
3η0 F0 +σ0
u2(t) =r− Ric= ∂logF
∂t = u′(
t) u(t)
. It implies that
uu′−u2r=−(n−1)3η0
F0 +σ0, u(0) = 1
which is an ordinary differential equation and has a solution as follow:
u2(t) =n−1 r
3η0
F0 +σ0(1−e
2rt) +e2rt
therefore
Ft2=
n−1 r
3η0
F0 +σ0(1−e
2rt) +e2rt
F02
and dλ
dt is obtained from (35).
Example 4.4. In this example we determine the behavior of the evolving spectrum on the Ricci solitons.
LetFtis a solution of the Ricci flow ∂logF∂t =−Ric. Ifϕia a time-independent
isometry such that
F(x, y) =ϕ∗
Ft(x, y)
is a solution of the un-normalized Ricci flow, because of ∂t∂logFt(x, y) =−Ric(x, y), F(0) =
F0,
∂
∂tlogF(x, y) = ∂ ∂tϕ
∗
logFt(x, y) =
∂
∂tlogFt(ϕ(x), ϕ∗(y))
since ϕis a isometry we have
∂
∂tlogF(x, y) = ∂
∂tlogFt(x, y) =−Ric(x, y) =−Ric(ϕ(x), ϕ∗(y)) =−Ric(F)
Definition 4.5. Let (M, Ft) is a solution of the Ricci flow and ϕt is a family of
diffeomorphisms. We says F(t)is Ricci soliton, when satisfies in
Let(M, F)and(M , F)be two closed Finsler manifolds and
ϕ: (M, g)→(M , F)
an isometry, then for p= 2 we have
e
g∆
p◦ϕ∗=ϕ∗◦eg∆p.
Therefore given a diffeomorphism ϕ:M →M we have that
ϕ: (SM, ϕ∗eg)→(SM,eg)
is an isometry, hence we conclude that (SM, ϕ∗
e
g), and (SM,eg) have the same spectrum
Specp(eg) =Specp(ϕ∗eg)
with eigenfunction fk andϕ∗fk respectively. If g(t)is a Ricci soliton on (Mn, g0)
then
Specp(eg(t)) =
1
u(t)Specp(g0e)
so that λ(t)satisfies
λ(t) = 1 u(t),
dλ dt =−
u′(
t) (u(t))2.
Example 4.6. Suppose that
Rn+=
(x1, x2, ..., xn)∈Rn|xi>0, i= 1, ..., n
has the metric
gij(x, y) =
φt(y) (xi)
2(p+1)
p
if i=j
0 if i6=j
(46)
where it is a solution for the un-normalized Ricci flow andφt(y)is a strictly positive
C∞
homogeneous function of degree zero andp >0. We use the formula
gij =
(xi)2(pp+1)
φt(y) if i=j 0 if i6=j
(47)
fori= 1, ..., n, we have
Gi=−p+ 1
2p (yi)2
xi
and
F2Ric(g(t)) = 0
thereforeRic(g(t)) = 0and the un-normalized Ricci flow equation implies that
∂g(t) ∂t = 0
174
Example 4.7. Suppose that R2 has the metric
gij(x, y) =φt(y)
4(x1)2+ 1 −2x2
−2x1 1
(48)
which is a solution for the the un-normalized Ricci flow, where φt(y) is a strictly
positiveC∞ homogeneous function of degree zero. We obtain
G1= 0, G2=−(y1)2
and
F2Ric(g(t)) = 0
thereforeRic(g(t)) = 0and the un-normalized Ricci flow equation implies that
∂g(t) ∂t = 0
hence, g(t) =g0 andλ(t) =λ(0).
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