EIGENVALUES VARIATION OF THE P-LAPLACIAN UNDER THE RICCI FLOW ON SM | Azami | Journal of the Indonesian Mathematical Society 215 858 1 PB
J. Indones. Math. Soc.
Vol. 22, No. 2 (2016), pp. 157–176.
EIGENVALUES VARIATION OF THE P-LAPLACIAN
UNDER THE RICCI FLOW ON SM
Shahroud Azami1 and Asadollah Razavi2
1
Department of Mathematics, Faculty of Science, Imam khomeini
international university, Qazvin, Iran
azami@sci.ikiu.ac.ir
2
Department 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 the p-laplace operator for
Sasakian metric acting on the space of functions on SM . We find the first variation
formula for the eigenvalues of p-Laplacian on SM evolving by the Ricci flow on M
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
operator p-Laplace untuk metrik Sasakian yang berlaku pada ruang fungsi di SM .
Kami memperoleh rumus variasi pertama dan memberikan beberapa contoh untuk
nilai eigen dari p-Laplacian pada SM yang melibatkan aliran Ricci pada M .
Kata kunci: aliran Ricci, manifold Finsler, operator p-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.
157
S. Azami and A. Razavi
158
Analytic Function. Recently, there are many mathematicians who have investigated
properties of the eigenvalues of p-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 mathematics 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 transformations and this property is preserved by the flow.
Let M be a manifold with a Finsler metric g0 (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
= −Ric,
∂t
(1)
with the initial condition
F (0) = F0
or equivalently satisfies the equations
∂gij
= −2Ricij , g(0) = g0
∂t
(2)
where Ric is the Ricci tensor of g(t), Ricij = ( 12 F 2 Ric)yi yj . 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, when M is compact.
One often considers the normalized Ricci flow
Z
1
∂logF
= −Ric +
Ric dv, F (0) = F0 .
(3)
∂t
vol(SM ) SM
or
∂gij
2
= −2Ricij +
∂t
vol(SM )
Z
Ric dvgij , g(0) = g0
(4)
SM
Under this normalized flow, the volume of the solution metrics remains constant in
time. Short time exitance and uniqueness for solution to the Ricci flow on [0, T )
have been shown by Hamilton in [5] and by DeTurk in [7] for Riemannian manifolds
and by the authors in [1] for important Berwald manifolds.
Eigenvalues variation of the P -Laplacian
159
2. Preliminaries
Let M be an n-dimensional C ∞ manifold. For a point x ∈ M , denote by
Tx M the tangent space at x ∈ M ,and by T M = ∪x∈M Tx M the tangent bundle of
M . Any element of T M has the form (x, y), where x ∈ M and y ∈ Tx M .
Definition 2.1. A Finsler metric on a manifold M is a function F : T M0 → [0, ∞)
which has the following properties:
(i): F (x, αy) = αF (x, y), ∀α > 0;
(ii): F (x, y) is C ∞ on T M0 ;
(iii): For any non-zero tangent vector y ∈ Tx M , the associated quadratic form
gy : Tx M × Tx M → R on T M is an inner product, where
1 ∂2 2
.
F (x, y + su + rv)
gy (u, v) :=
2 ∂s∂r
s=r=0
The pair (M, F ) is called a Finsler manifold.
Let us denote by Sx M the set consisting of all rays [y] := {λy|λ > 0}, where
y ∈ Tx M0 . The Sphere bundle of M , i.e. SM , is the union of Sx M ’s :
SM = ∪ Sx M
x
SM has a natural (2n−1)-dimensional manifold structure. We denote the elements
of SM by (x, [y]) where y ∈ Tx M0 . If there is not any confusion we write (x, y) for
∂2F 2
(x, [y]). In a local coordinate system xi , y i we have gij (x, y) = 21 ∂y
i ∂y j (x, y) and
ij
−1
(g ) := (gij ) . The geodesics of F are characterized locally by
dx
d2 xi
+ 2Gi (x, ) = 0
dt2
dt
where
Gi =
∂gjk
1 il
∂gjl
g
2 k −
yj yk .
4
∂x
∂xl
(5)
Definition 2.2. The coefficients of the Riemann curvature Ry = Rki dxi ⊗
given by
2 i
∂Gi
∂ 2 Gi j
∂Gi ∂Gj
j ∂ G
Rik := 2 k −
y
+
2G
−
∂x
∂xj ∂y k
∂y j ∂y k
∂y j ∂y k
i
∂R
∂Ri
∂ 2 Ri
∂ 2 Ri
and Rijk := 31 ∂ykj − ∂yjk , Rji kl := 13 ∂yj ∂ykl − ∂yj ∂ylk .
The Ricci scalar function of F is given by Ric :=
Ricci scalar is the Ricci tensor
1 2
F Ric
.
Ricij :=
2
yi yj
1
i
F 2 Ri .
∂
∂xi
are
(6)
A companion of the
(7)
S. Azami and A. Razavi
160
Definition 2.3. A Finsler metric is said to be an Einstain metric if the Ricci
scalar function is a function of x alone,equivalently Ricij = R(x)gij (see [14, 19]).
Definition 2.4. Let (M, F ) be a Finsler manifold, the Sasakian metric ge of g on
T M0 is defined as
δy i
δy j
ge = gij dxi ⊗ dxj + gij
⊗
(8)
F
F
∂
δ
then ge is a Riemannian metric on T M0 and { δxi , F ∂yi } is a coordinate base on
T M0 , where δxδ i =
δy i = dy i + Gij dxj .
∂
∂xi
i
δ
∂
− Gji ∂y∂ j and {dxi , δy
F } is the dual of { δxi , F ∂y i } where
e on T M0 with respect to the Sasakian
Remark. The Levi-Civita connection ∇
metric ge is locally expressed as follows:
δ
1
∂
e δ δ
∇
= −(Cikj + Rkij ) k + Fikj k
(9)
δxj δxi
2
∂y
δx
∂
δ
e ∂ ∂
= Cikj k − gih (Fjhk − Ghj k )g hk k
∇
i
j
∂y ∂y
∂y
δx
∂
1
δ
∂
e δ
= Fikj k + (Cikj + gih Rhlj g lk ) k
∇
δxj ∂y i
∂y
2
δx
e ∂ δ + Gkij ∂ ,
= ∇
∂y i δxj
∂y k
where
Cikj =
∂Gkj
1 kh ∂gij
1 kh δghi
δghj
δgij
k
k
g
,
F
=
g
(
+
−
),
G
=
,
i
j
i
j
2
∂y h
2
δxj
δxi
δxh
∂y i
Rkij =
δGkj
δ
δ
∂
δ
∂
∂
δGki
−
, [ i , j ] = Rkij k , [ i , j ] = Gkij k .
j
i
δx
δx
δx δx
∂y
δx ∂y
∂y
and
Lemma 2.5. For a Sasakian metric ge and any f : T M → R, there exists a unique
vector field Y ∈ X (T M ) such that
where
and X1i , X2i are C ∞
e = df (X),
e ∀X
e ∈ X (T M )
ge(Y, X)
(10)
e = Xi δ + XiF ∂
X
1
2
δxi
∂y i
function on T M . Here we take Y = 0 if df = 0.
˜ . We call ∇f
˜ the gradient of f and
Denote the vector field Y in (10 ) by ∇f
˜
define the divergence div X as follows:
˜ = tr∇
˜X
˜
div X
Eigenvalues variation of the P -Laplacian
161
Definition 2.6. According to the above definition, the gradient of a function f is
˜ = g ij δf δ + F 2 g ij ∂f ∂
∇f
δxi δxj
∂y i ∂y j
(11)
˜ with respect to the Riemannian metric ge is given
therefore, the norm of ∇f
by
˜ |2 = ge(∇f,
e ∇f
e ) = g ij δf δf + F 2 g ij ∂f ∂f .
|∇f
δxi δxj
∂y i ∂y j
(12)
Definition 2.7. Let M be a compact Finsler manifold. The Laplace operator of f
on T M is defined as follows:
∆f
=
+
−
∂Grj ∂f
∂2f
∂2f
∂2f
∂2f
−
− Grj i r − Gsi s j + Grj Gsi s r )
i
j
i
r
∂x ∂x
∂x ∂y
∂x ∂y
∂y ∂x
∂y ∂y
2
1
δf
δf
∂ f
(13)
g ij F 2 i j + g ij (Cikj + Rkij ) k − F g ij Fikj k
∂y ∂y
2
δx
δx
∂f
δf
F g ij Cikj k − F 2 g ij gih (Gjhl − Fjhl )g lk k .
∂y
δy
g ij (
Definition 2.8. Let M be a compact Finsler manifold. The p-Laplace operator of
f : SM → R, f ∈ W 1,p (SM ) for 1 < p < ∞ is defined as follows:
△p f
=
=
where
e |p−2 ∇f
e )
div(|∇f
e |p−2 ∆f + (p − 2)|∇f
e |p−4 (Hessf )(∇f,
e ∇f
e )
|∇f
(14)
e ∇f
e )(X, Y ) = Y.(X.f ) − (∇
e Y X).f, X, Y ∈ X (SM )
(Hessf )(X, Y ) = ∇(
and in local coordinate, we have:
e kij ∂k f.
(Hessf )(∂i , ∂j ) = ∂i ∂j f − Γ
Note. If f is a function of x alone, or suppose that is the lifting of f : M → R
then
2
∂ f
e kij ∂f
−
Γ
(15)
∆f = g ij
∂xi ∂xj
∂xk
e
e k is christoffel symbol of ∇.
where Γ
ij
2.1. Eigenvalues of the p-Laplacian.
Definition 2.9. Let (M n , F ) be a compact Finsler manifold and f : SM → R. We
say that λ is an eigenvalue of the p-Laplace operator whenever
g
e
∆p f + λ|f |p−2 f = 0
(16)
then f is said to be the eigenfunction associated to λ, or equivalently they satisfy
in
Z
Z
p−2
e
e
e
|f |p−2 f ϕdv
∀ϕ ∈ W01,2 (SM )
(17)
|∇f |
< ∇f, ∇ϕ > dv = λ
SM
SM
S. Azami and A. Razavi
162
where W01,p (SM ) is closure of C0∞ (SM ) in Sobolev W 1,p (SM ).
By substitution ϕ = f in (17) we have:
R
e |p dv
|∇f
λ = RSM
.
|f |p dv
SM
Normalized eigenfunctions are defined as follows:
Z
Z
f |f |p−2 dv = 0,
|f |p dv = 1.
SM
(18)
(19)
SM
Suppose that (M n , Ft ) is a solution of the Ricci flow on the smooth manifold
(M n , F0 ) in the interval [0, T ) and
Z
e (x, y)|p dvt
λ(t) =
|∇f
(20)
SM
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) on M n
Specp (e
g ) = {0 = λ0 (g) ≤ λ1 (g) ≤ λ2 (g) ≤ ... ≤ λk (g) ≤ ...}
is the spectrum of ∆p = ge∆p . In what follows we assume the existence and C 1 differentiability of the elements λ(t) and f (t), under a Ricci flow deformation g(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 for λ(t) under the
Ricci flow. Let (M n , Ft ), t ∈ [0, T ),be a deformation of Finsler metric F0 . Assume
that λ(t) is the eigenvalue of ∆p , f = f (x, y, t) satisfies
∆p f + λ|f |p−2 f = 0
and
R
SM
|f |p dv = 1, using (12), we have:
d e 2
|∇f |
dt
=
+
+
∂ ij δf δf
∂ δf δf
(g ) i j + g ij ( i ) j
∂t
δx δx
∂t δx δx
δf ∂ δf
∂(F 2 ) ij ∂f ∂f
g ij i ( j ) +
g
δx ∂t δx
∂t
∂y i ∂y j
∂
∂f ∂f
∂f ′ ∂f
F 2 (g ij ) i j + 2F 2 g ij i j .
∂t
∂y ∂y
∂y ∂y
(21)
where
∂ ij
∂
(g ) = −g il g jk (glk )
∂t
∂t
(22)
Eigenvalues variation of the P -Laplacian
163
and
∂ δf
(
)
∂t δxi
∂ ∂f
r ∂f
−
G
i
∂t ∂xi
∂y r
∂f ′
∂f ′
∂
∂f
− Gri r − (Gri ) r
i
∂x
∂y
∂t
∂y
δf ′
∂
∂f
− (Gri ) r
i
δx
∂t
∂y
=
=
=
(23)
therefore, a substitution of (22) and (23) in (21), implies that:
Proposition 3.1. Let (M n , Ft )) be a deformation of Finsler manifold (M n , F0 ), then
d e p
|∇f |
dt
∂
p e p−2
δf δf
δf ′ δf
−g il g jk (glk ) i j + 2g ij i j
|∇f |
2
∂t
δx δx
δx δx
∂f δf
∂F ij ∂f ∂f
∂
p e p−2
|
g
2F
−2g ij (Gri ) r j + |∇f
∂t
∂y δx
2
∂t
∂y i ∂y j
∂f ∂f
∂f ′ ∂f
∂
−F 2 g il g jk (glk ) i j + 2F 2 g ij i j .✷
∂t
∂y ∂y
∂y ∂y
=
On the other hand we have
∂
∂
d
(dv) = g ij (gij ) − n (logF ) dv.
dt
∂t
∂t
(24)
Now, we get the following two integrability conditions:
0=
d
dt
Z
|f |p−2 f dv = (p − 1)
SM
Z
|f |p−2 f ′ dv +
SM
Z
|f |p−2 f
SM
d
dv
dt
therefore
(p − 1)
Z
|f |
p−2 ′
f dv = −
SM
Z
|f |
p−2
SM
f
∂
∂
(gij ) − n (logF ) dv
g
∂t
∂t
ij
(25)
and
0=
d
dt
Z
|f |p dv = p
SM
Z
f f ′ |f |p−2 dv +
SM
Z
|f |p
SM
d
dv
dt
which implies
p
Z
′
f f |f |
SM
p−2
∂
ij ∂
(gij ) − n (logF ) dv.
dv = −
|f | g
∂t
∂t
SM
Z
p
(26)
S. Azami and A. Razavi
164
Now, if we suppose that g(t) is a solution of the un-normalized Ricci flow (1) and
(2), then we have:
Z
Z
dλ
d e p
e |p d (dv)
=
( |∇f | )dv +
|∇f
dt
dt
dt
SM
SM
Z
p
δf δf
δf ′ δf
=
−g il g jk (−2Riclk ) i j + 2g ij i j
2 SM
δx δx
δx δx
Z
p
∂f ∂f
r ∂f δf
p−2
ij ∂
e
(G )
|∇f | dv +
2F 2 (−Ric)g ij i j
−2g
∂t i ∂y r δxj
2 SM
∂y ∂y
′
∂f ∂f
2 il jk
2 ij ∂f ∂f
e |p−2 dv
−F g g (−2Riclk ) i j + 2F g
|∇f
∂y ∂y
∂y i ∂y j
Z
e |p g ij (−2Ricij ) − n(−Ric) dv
|∇f
+
Z SM
Z
e ∇f
e )|∇f
e |p−2 dv + p
e ′ , ∇f
e )|∇f
e |p−2 dv
= p
Ric(∇f,
ge(∇f
SM
SM
Z
e |p −2g ij Ricij + nRic dv
|∇f
+
SM
Z
∂f δf e p−2
d
| dv
g ij (Gri ) r j |∇f
−p
dt
∂y
δx
SM
Z
∂f ∂f e p−2
RicF 2 g ij i j |∇f
−p
| dv
∂y ∂y
SM
where
d
r
dt (Gi )
Hence
d
(Gri )
dt
is obtained as follows:
∂gjk
1 rl
∂gjl
r
G = g
2 k −
yj yk ,
4
∂x
∂xl
=
Gri =
∂Gr
∂y i
∂(gab )
1
∂
∂gjl
∂gjk
2
yj yk
− g rα g aβ (gαβ )g lb
−
4
∂t
∂y i
∂xk
∂xl
∂(gab )
1
∂gjl
∂
∂gjk
− g lα g bβ (gαβ )g ra
2
yj yk
−
4
∂t
∂y i
∂xk
∂xl
′
)
∂gjl
∂gjk
1 ra lb ∂(gab
2 k −
yj yk
− g g
4
∂y i
∂x
∂xl
′
′
∂gjl
∂gjk
∂(gab )
1
− g ra g lb
2
−
yj yk
(27)
4
∂y i
∂xk
∂xl
∂(gαβ )
∂ 2 gjl
∂ 2 gjk
1
yj yk
2 i k −
+ g rα g lβ
4
∂t
∂y ∂x
∂y i ∂xl
(
)
′
′
∂ 2 gjl
∂ 2 gjk
1
+ g rl 2 i k −
yj yk
4
∂y ∂x
∂y i ∂xl
′
′
∂(gαβ )
1 rl
∂gil
∂gil
∂gik
∂gik
1
k
y
+
yk
2 k −
2
g
−
+ g rα g lβ
4
∂t
∂x
∂xl
4
∂xk
∂xl
′
′
∂gjl
∂gij
∂(gαβ )
1 rl
∂gjl
∂gij
1
j
y
+
yj .
2 i −
2
g
−
+ g rα g lβ
4
∂t
∂x
∂xl
4
∂xi
∂xl
Eigenvalues variation of the P -Laplacian
165
From the un-normalized Ricci flow, we can then write
d
(Gri )
dt
=
∂(gab )
1 rα aβ
∂gjl
∂gjk
2
yj yk
g g Ricαβ g lb
−
2
∂y i
∂xk
∂xl
∂(gab )
∂gjl
∂gjk
1
2
yj yk
−
+ g lα g bβ Ricαβ g ra
2
∂y i
∂xk
∂xl
∂(Ricab )
∂gjl
∂gjk
1
2
yj yk
−
+ g ra g lb
2
∂y i
∂xk
∂xl
∂(gab )
1
∂Ricjl
∂Ricjk
+ g ra g lb
2
yj yk
−
2
∂y i
∂xk
∂xl
∂ 2 gjl
∂ 2 gjk
1
yj yk
− g rα g lβ Ricαβ 2 i k −
2
∂y ∂x
∂y i ∂xl
2
∂ Ricjl
∂ 2 Ricjk
1
yj yk
− g rl 2 i k −
2
∂y ∂x
∂y i ∂xl
1
∂gil
∂gik
− g rα g lβ Ricαβ 2 k −
yk
2
∂x
∂xl
∂Ricik
∂Ricil
1 rl
−
yk
2
− g
2
∂xk
∂xl
∂gjl
∂gij
1
yj
− g rα g lβ Ricαβ 2 i −
2
∂x
∂xl
1
∂Ricij
∂Ricjl
− g rl 2
yj .
−
2
∂xi
∂xl
Using (26) we obtain
Z
e ′ , ∇f
e )|∇f
e |p−2 dv
p
ge(∇f
=
SM
=
pλ
Z
−λ
(28)
f ′ f |f |p−2 dv
SM
Z
SM
We have thus proved the following proposition:
|f |p −2g ij Ricij + nRic dv.
Proposition 3.2. Let (M n , Ft )) be a solution of the un-normalized Ricci flow on the
smooth Finsler manifold (M n , F0 ). If λ(t) denotes the evolution of an eigenvalue
under the Ricci flow, then
Z
dλ
e ∇f
e )|∇f
e |p−2 dv
Ric(∇f,
= p
dt
SM
Z
e |p ) 2g ij Ricij − nRic dv
+
(λ|f |p − |∇f
(29)
SM
Z
∂
∂f δf e p−2
g ij (Gri ) r j |∇f
−p
| dv
∂t
∂y δx
ZSM
∂f ∂f e p−2
−p
Ric F 2 g ij i j |∇f
| dv
(30)
∂y ∂y
SM
where f is the associated normalized evolving eigenfunction.✷
S. Azami and A. Razavi
166
Note. Let f : SM → R be a lifting of f : M → R. We have:
Z
dλ
e ∇f
e )|∇f
e |p−2 dv
= p
Ric(∇f,
dt
SM
Z
e |p ) −2g ij Ricij + nRic dv
(λ|f |p − |∇f
+
SM
ij
and in this case, if −2g Ricij + nRic is a constant, then
Z
dλ
e ∇f
e )|∇f
e |p−2 dv.
=p
Ric(∇f,
dt
SM
Corollary 3.3. Let (M n , Ft ) be a solution of the un-normalized Ricci flow on the
smooth Riemannian manifold (M n , F0 ), i.e. Ft , F0 are Riemannian metric. If λ(t)
denotes the evolution of an eigenvalue under Ricci flow, then:
Z
Z
dλ
p−2
e |p )dv
e
e
e
R(λ|f |p − |∇f
Ric(∇f, ∇f )|∇f | dv +
= p
dt
SM
SM
Z
∂f δf e p−2
∂
| dv
(31)
−p
g ij (Gir ) r j |∇f
∂t
∂y
δx
SM
Z
∂f ∂f e p−2
p
| dv
RF 2 g ij i j |∇f
−
n SM
∂y ∂y
where R is the scalar curvature of M .
Proof :
If Ft is the Riemannian metric, then
Ric =
1
R,
n
(32)
and
2g ij Ricij − nRic = R,
therefore (31) is obtained by replacing (32) and (33) in (29).
✷
(33)
Corollary 3.4. Let (M 2 , Ft ) be a solution of the un-normalized Ricci flow on the
smooth Riemannian surface (M 2 , F0 ). If λ(t) denotes the evolution of an eigenvalue
under the Ricci flow, then:
Z
Z
p
dλ
p
e |p )dv
e
R(λ|f |p − |∇f
=
R|∇f | dv +
dt
2 SM
SM
Z
∂
∂f δf e p−2
| dv
−p
g ij (Gir ) r j |∇f
∂t
∂y
δx
SM
Z
∂f ∂f e p−2
p
| dv
RF 2 g ij i j |∇f
−
n SM
∂y ∂y
where R is the scalar curvature of M .
Proof : In dimension n = 2, for a Riemannian manifold, we have:
1
Ric = Rg,
2
(34)
Eigenvalues variation of the P -Laplacian
167
hence the corollary is obtained by replacing (34) in (29). ✷
Corollary 3.5. Let (M n , Ft ) be a solution of the un-normalized Ricci flow on the
smooth homogenous Riemannian manifold (M n , F0 ). If λ(t) denotes the evolution
of an eigenvalue under the Ricci flow, then:
dλ
dt
=
p
Z
SM
Z
e ∇f
e )|∇f
e |p−2 dv
Ric(∇f,
∂f δf e p−2
∂
(Gir ) r j |∇f
| dv
∂t
∂y
δx
SM
Z
∂f ∂f e p−2
p
F 2 g ij i j |∇f
| dv
−R
n SM
∂y ∂y
−p
g ij
where R is the scalar curvature of M .
Proof :
Since the evolving metric remains homogenous and a Riemannian homogenous 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 (M n , Ft ) be a solution of the normalized Ricci flow on the
smooth Finsler manifold (M n , F0 ). If λ(t) denotes the evolution of an eigenvalue
under Ricci flow, then:
dλ
dt
=
Z
e ∇f
e )|∇f
e |p−2 dv
Ric(∇f,
(n − p)rλ + p
SM
Z
e |p ) 2g ij Ricij − nRic dv
+
(λ|f |p − |∇f
SM
Z
∂f δf e p−2
∂
g ij (Gsi ) s j |∇f
−p
| dv
∂t
∂y δx
SM
Z
∂f ∂f e p−2
−p
F 2 (Ric − r) g ij i j |∇f
| dv
∂y ∂y
SM
where f is the associated normalized evolving eigenfunction, r =
Ric dv
vol(SM ) .
SM
Proof: In the normalized case, the integrability conditions read as follows
Z
Z
′
p−2
(36)
|f |p 2g ij Ricij − nr − nRic dv.
f f |f | dv =
p
SM
Since
R
(35)
SM
d
(dv) = −2g ij Ricij + nr + nRic dv
dt
(37)
S. Azami and A. Razavi
168
using (24), (27) and the above equation, we can then write
dλ
dt
=
=
Z
Z
d
d
e |p
e |p )dv +
|∇f
|∇f
(dvt )
dt
dt
SM
SM
)
(
Z
′
p
δf
δf δf
ij δf
il jk
e |p−2 dv
|∇f
+ 2g
−g g (−2Riclk + 2rglk )
i
j
i
j
2 SM
δx δx
δx δx
Z
p
2
ij ∂f ∂f
+
2F (−Ric + r)g
2 SM
∂y i ∂y j
∂f ∂f
2 il jk
e |p−2 dv
|∇f
−F g g (−2Riclk + 2rglk )
∂y i ∂y j
)
(
Z
′
∂f
p
ij ∂
2 ij ∂f
r ∂f δf
e |p−2 dv
|∇f
−2g
+ 2F g
(Gi )
+
r
j
i
j
2 SM
∂t
∂y δx
∂y ∂y
Z
n
o
e |p −2g ij Ricij + nr + nRic dv
|∇f
+
(
SM
=
(n − p)rλ + p
+
Z
−p
SM
Z
e |
|∇f
g
SM
ij
p
Z
n
d
dt
SM
−2g
r
e ∇f
e )|∇f
e |p−2 dv + p
Ric(∇f,
ij
(Gi )
o
Ricij + nRic dv
∂f
δf
∂y r δxj
Z
e |
|∇f
p−2
Z
SM
(38)
e ′ , ∇f
e )|∇f
e |p−2 dv
g
e(∇f
dv
p−2
2
ij ∂f ∂f e
|∇f |
dv
F (Ric − r) g
−p
∂y i ∂y j
SM
but
p
Z
SM
e ′ , ∇f
e )|∇f
e |p−2 dv
ge(∇f
=
=
Z
f ′ f |f |p−2 dv
pλ
Z SM
λ
|f |p 2g ij Ricij − nr − nRic dv (39)
SM
∂
′
and ∂t
(Gsi ) is obtained by replacing F ′ and gij
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 (M n , Ft ) be a solution of the normalized Ricci flow on the
smooth Riemannian manifold (M n , F0 ). If λ(t) denotes the evolution of an eigenvalue under Ricci flow, then:
Z
dλ
e ∇f
e )|∇f
e |p−2 dv
= (n − p)rλ + p
Ric(∇f,
dt
SM
Z
Z
∂
∂f δf e p−2
p
p
e
g ij (Gsi ) s j |∇f
R(λ|f | − |∇f | )dv − p
+
| dv (40)
∂t
∂y δx
SM
SM
Z
∂f ∂f e p−2
1
| dv
−p
( R − r)F 2 g ij i j |∇f
n
∂y
∂y
SM
where R is the scalar curvature of M .
Corollary 3.8. Let (M 2 , Ft ) be a solution of the normalized Ricci flow on the
smooth Riemannian surface (M 2 , F0 ). If λ(t) denotes the evolution of an eigenvalue
Eigenvalues variation of the P -Laplacian
169
under Ricci flow, then:
Z
p
dλ
e |p dv
= (2 − p)rλ +
R|∇f
dt
2 SM
Z
Z
∂
∂f δf e p−2
e |p )dv − p
g ij (Gsi ) s j |∇f
R(λ|f |p − |∇f
+
| dv
∂t
∂y δx
SM
SM
Z
∂f ∂f e p−2
R
| dv
−p
( − r)F 2 g ij i j |∇f
∂y ∂y
SM 2
where R is the scalar curvature of M .
Corollary 3.9. Let (M n , Ft ) be a solution of the normalized Ricci flow on the
smooth homogenous Riemannian manifold (M n , F0 ). If λ(t) denotes the evolution
of an eigenvalue under Ricci flow, then:
Z
dλ
e ∇f
e )|∇f
e |p−2 dv
Ric(∇f,
= (n − p)rλ + p
dt
SM
Z
∂f δf e p−2
∂
| dv
−p
g ij (Gsi ) s j |∇f
∂t
∂y
δx
ZSM
1
∂f ∂f e p−2
( R − r)F 2 g ij i j |∇f
−p
| dv
∂y ∂y
SM n
where R is the scalar curvature of M .
4. Examples
In this section, we will find the variational formula for some of Finsler manifolds.
Example 4.1. Let (M n , F0 ) be an Einstein manifold i.e. there exists a constant
a such that Ric(F0 ) = aF02 . 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
where u(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
a
g(t),
Ric(g(t)) = Ric(g0 ) = ag0 =
1 − 2at
a
1
Ric(g0 ) =
,
Ric(gt ) =
1 − 2at
1 − 2at
Ft2 = (1 − 2at)F02
therefore
2g ij Ricij − nRic =
an
1 − 2at
S. Azami and A. Razavi
170
and
Also
e ∇f
e )=
Ric(∇f,
Gi (t)
=
=
a
a
e ∇f
e )=
e |2 .
ge(∇f,
|∇f
1 − 2at
1 − 2at
1 il ∂gjl
∂gjk j k
g {2 k −
}y y
4
∂x
∂xl
∂(g0 )jk j k
∂(g0 )jl
1
−
}y y = Gi (0)
(g0 )il {2
4
∂xk
∂xl
therefore
∂
(Gi ) = 0.
∂t r
Using the un-normalized Ricci flow equation (2) and (29) ,we obtain the following
relation:
Z
Z
a
an
dλ
e |p dv + λ
|f |p
= p
|∇f
dv
dt
1 − 2at
1 − 2at
Z M
Z M
an
∂f ∂f e p−2
a
e |p dv − p
|∇f
F 2 g ij i j |∇f
| dv
−
1
−
2at
1
−
2at
∂y
∂y
M
SM
Z
pa
∂f ∂f e p−2
=
λ−
| dv
F 2 g ij i j |∇f
1 − 2at
∂y
∂y
SM
Now, If we suppose that
R gt = u(t)g0 , u(0) = 1 is a solution of the normalized Ricci
1
flow and r = V ol(SM
) SM Ricdv, then from (3) we have
u′ g0 = −2ag0 + 2rug0 .
It implies that
a
(1 − e2rt ) + e2rt .
r
o
na
(1 − e2rt ) + e2rt
g(t) =
r
u(t) =
So that we have:
and
−1
a
2rt
2rt
Ric(g(t)) = a
g0
(1 − e ) + e
r
−1
a
(1 − e2rt ) + e2rt
Ric(g(t)) = a
r
−1
a
ij
2rt
2rt
2g Ricij − Ricij = an
(1 − e ) + e
,
r
also
Gi (t) = Gi (0),
therefore
∂Gri
= 0.
∂t
Eigenvalues variation of the P -Laplacian
171
Using (35), we obtain the following:
−1
a
dλ
2rt
2rt
= (n − p)rλ + paλ
(1 − e ) + e
dt
r
−1
Z
a
∂f 2 e p−2
− r (y i
) |∇f | dv
− p
a
(1 − e2rt ) + e2rt
r
∂yi
SM
−1
a
2rt
2rt
=
(n − p)r + pa
(1 − e ) + e
λ
r
Z
−1
∂f ∂f e p−2
a
F 2 g ij i j |∇f
−r
(1 − e2rt ) + e2rt
| dv.
− p a
r
∂y ∂y
SM
Remark. Let (M n , F0 ) be a Finsler manifold of dimension n ≥ 3. Suppose that the
flag curvature k = k(x) is isotropic and a function of x ∈ M alone then k = constant
and therefore (M n , 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
3η
Ric = (n − 1){ + σ}F 2
F
where η is a 1-form and σ = σ(x) is scalar function.
Example 4.3. If we suppose that Ft = u(t)F0 , u(0) = 1 is a solution of the Ricci
flow, then:
Ric(Ft ) =
2
0
0
(n − 1){ 3η
(n − 1){ 3η
Ric(F0 )
Ric(Ft )
F0 + σ0 }F0
F0 + σ0 }
=
=
=
Ft2
(u(t))2 F02
(u(t))2 F02
(u(t))2
Now the Ricci flow (1) implies that
−
0
(n − 1){ 3η
F0 + σ0 }
(u(t))2
= −Ric =
∂logF
F′
u′ (t)
= t =
∂t
Ft
u(t)
or equivalently
uu′ = (n − 1){
3η0
+ σ0 }
F0
By integration we have:
u2 (t) = −2(n − 1){
3η0
+ σ0 }t + c
F0
with condition u(0) = 1 we have:
u2 (t) = 1 − 2(n − 1){
therefore
Ft2
=
3η0
+ σ0 }t
F0
3η0
1 − 2(n − 1){
+ σ0 }t F02
F0
(41)
S. Azami and A. Razavi
172
and
e ∇f
e ) = 1 (n − 1)g il g js (3η0 F0 )yl ys + 2σ0 (g0 )ls ∇
e if ∇
e j f.
(42)
Ric(∇f,
2
Also we have
0
(n − 1){ 3η
F0 + σ0 }
.
2g ij Ricij −nRic = (n−1)g ij (3η0 F0 )yi yj + 2σ0 (g0 )ij −n
0
1 − 2(n − 1){ 3η
F0 + σ0 }t
(43)
Note that
∂η ∂F
(44)
gij (t) = (1 − 2(n − 1)σ0 t)(g0 )ij − 6(n − 1)t i j .
∂y ∂y
By replacing (42),(43) and (44) in (29) we obtain the variation of an eigenvalue.
Now, if we suppose that
R Ft = u(t)F0 , u(0) = 1 is a solution of the normalized Ricci
1
flow and r = V ol(SM
) SM Ric dv, then from (3), we have:
r−
0
(n − 1) 3η
F0 + σ0
u2 (t)
= r − Ric =
∂logF
u′ (t)
=
∂t
u(t)
. It implies that
uu′ − u2 r = −(n − 1)
3η0
+ σ0 ,
F0
u(0) = 1
which is an ordinary differential equation and has a solution as follow:
n − 1 3η0
u2 (t) =
+ σ0 (1 − e2rt ) + e2rt
r F0
therefore
n − 1 3η0
Ft2 =
+ σ0 (1 − e2rt ) + e2rt F02
r F0
is
obtained
from
(35).
and dλ
dt
Example 4.4. In this example we determine the behavior of the evolving spectrum
on the Ricci solitons.
Let Ft is a solution of the Ricci flow ∂logF
= −Ric. If ϕ ia a time-independent
∂t
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 ,
∂
∂
∂
logF (x, y) = ϕ∗ logFt (x, y) = logFt (ϕ(x), ϕ∗ (y))
∂t
∂t
∂t
since ϕ is a isometry we have
∂
∂
logF (x, y) = logFt (x, y) = −Ric(x, y) = −Ric(ϕ(x), ϕ∗ (y)) = −Ric(F )
∂t
∂t
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
Ft2 = u(t)ϕ∗t F02 .
(45)
Eigenvalues variation of the P -Laplacian
173
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
g
e
e
∆ p ◦ ϕ∗ = ϕ ∗ ◦ g ∆ p .
Therefore given a diffeomorphism ϕ : M → M we have that
ϕ : (SM, ϕ∗ ge) → (SM, ge)
is an isometry, hence we conclude that (SM, ϕ∗ ge), and (SM, ge) have the same
spectrum
Specp (e
g ) = Specp (ϕ∗ ge)
with eigenfunction fk and ϕ∗ fk respectively. If g(t) is a Ricci soliton on (M n , g0 )
then
1
Specp (e
g0 )
Specp (e
g (t)) =
u(t)
so that λ(t) satisfies
dλ
u′ (t)
1
,
=−
.
u(t) dt
(u(t))2
λ(t) =
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
0
if i = j
(46)
if i 6= j
where it is a solution for the un-normalized Ricci flow and φt (y) is a strictly positive
C ∞ homogeneous function of degree zero and p > 0. We use the formula
(xi ) 2(p+1)
p
if i = j
ij
φt (y)
g =
(47)
0
if i 6= j
for i = 1, ..., n, we have
Gi = −
p + 1 (y i )2
2p xi
and
F 2 Ric(g(t)) = 0
therefore Ric(g(t)) = 0 and the un-normalized Ricci flow equation implies that
∂g(t)
=0
∂t
hence, g(t) = g0 and λ(t) = λ(0).
174
S. Azami and A. Razavi
Example 4.7. Suppose that R2 has the metric
gij (x, y) = φt (y)
4(x1 )2 + 1
−2x1
−2x2
1
(48)
which is a solution for the the un-normalized Ricci flow, where φt (y) is a strictly
positive C ∞ homogeneous function of degree zero. We obtain
G1 = 0, G2 = −(y 1 )2
and
F 2 Ric(g(t)) = 0
therefore Ric(g(t)) = 0 and the un-normalized Ricci flow equation implies that
∂g(t)
=0
∂t
hence, g(t) = g0 and λ(t) = λ(0).
References
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[3] Bartheleme, T., ”A natural Finsler-Laplace operator”, Israel J. Math. 196 (2013), 375-412.
[4] Cheng, Q. M., Yang, H. C., ” Estimates on eigenvalues of Laplacian”, Math. Ann. 331 (2005),
445-460.
[5] Chow, B., Knopf, D., The Ricci flow: An Introduction, Mathematical Surveys and Monographs, AMS, 2004.
[6] Chow, B., Lu, P., Ni, L., Hamilton’s Ricci flow, Vol. 77, AMS, 2006.
[7] DeTurck, D.M., ” Deforming metrics in the direction of their Ricci tensors”, J. Differential
Geometry, 18 (1983), 157-162.
[8] Friedan, D., Nonlinear Models in 2 + ǫ Dimensions, PhD, University of California, Berkeley,
USA, 1980.
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spectral geometry”, Comm. Partial Differential Equations, 19 (1994), 2037-2055.
[10] Headrick, M., Wiseman, T., ”Ricci Flow and Black Holes”, Class. Quantum Grav. 23 (2006),
6683-6707.
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59 (Kluwer Academic Publishers, Dordrecht, Boston, London, 1994).
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and applications”, Math. Appl., 459 (1998), 187-198.
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176
S. Azami and A. Razavi
Vol. 22, No. 2 (2016), pp. 157–176.
EIGENVALUES VARIATION OF THE P-LAPLACIAN
UNDER THE RICCI FLOW ON SM
Shahroud Azami1 and Asadollah Razavi2
1
Department of Mathematics, Faculty of Science, Imam khomeini
international university, Qazvin, Iran
azami@sci.ikiu.ac.ir
2
Department 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 the p-laplace operator for
Sasakian metric acting on the space of functions on SM . We find the first variation
formula for the eigenvalues of p-Laplacian on SM evolving by the Ricci flow on M
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
operator p-Laplace untuk metrik Sasakian yang berlaku pada ruang fungsi di SM .
Kami memperoleh rumus variasi pertama dan memberikan beberapa contoh untuk
nilai eigen dari p-Laplacian pada SM yang melibatkan aliran Ricci pada M .
Kata kunci: aliran Ricci, manifold Finsler, operator p-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.
157
S. Azami and A. Razavi
158
Analytic Function. Recently, there are many mathematicians who have investigated
properties of the eigenvalues of p-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 mathematics 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 transformations and this property is preserved by the flow.
Let M be a manifold with a Finsler metric g0 (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
= −Ric,
∂t
(1)
with the initial condition
F (0) = F0
or equivalently satisfies the equations
∂gij
= −2Ricij , g(0) = g0
∂t
(2)
where Ric is the Ricci tensor of g(t), Ricij = ( 12 F 2 Ric)yi yj . 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, when M is compact.
One often considers the normalized Ricci flow
Z
1
∂logF
= −Ric +
Ric dv, F (0) = F0 .
(3)
∂t
vol(SM ) SM
or
∂gij
2
= −2Ricij +
∂t
vol(SM )
Z
Ric dvgij , g(0) = g0
(4)
SM
Under this normalized flow, the volume of the solution metrics remains constant in
time. Short time exitance and uniqueness for solution to the Ricci flow on [0, T )
have been shown by Hamilton in [5] and by DeTurk in [7] for Riemannian manifolds
and by the authors in [1] for important Berwald manifolds.
Eigenvalues variation of the P -Laplacian
159
2. Preliminaries
Let M be an n-dimensional C ∞ manifold. For a point x ∈ M , denote by
Tx M the tangent space at x ∈ M ,and by T M = ∪x∈M Tx M the tangent bundle of
M . Any element of T M has the form (x, y), where x ∈ M and y ∈ Tx M .
Definition 2.1. A Finsler metric on a manifold M is a function F : T M0 → [0, ∞)
which has the following properties:
(i): F (x, αy) = αF (x, y), ∀α > 0;
(ii): F (x, y) is C ∞ on T M0 ;
(iii): For any non-zero tangent vector y ∈ Tx M , the associated quadratic form
gy : Tx M × Tx M → R on T M is an inner product, where
1 ∂2 2
.
F (x, y + su + rv)
gy (u, v) :=
2 ∂s∂r
s=r=0
The pair (M, F ) is called a Finsler manifold.
Let us denote by Sx M the set consisting of all rays [y] := {λy|λ > 0}, where
y ∈ Tx M0 . The Sphere bundle of M , i.e. SM , is the union of Sx M ’s :
SM = ∪ Sx M
x
SM has a natural (2n−1)-dimensional manifold structure. We denote the elements
of SM by (x, [y]) where y ∈ Tx M0 . If there is not any confusion we write (x, y) for
∂2F 2
(x, [y]). In a local coordinate system xi , y i we have gij (x, y) = 21 ∂y
i ∂y j (x, y) and
ij
−1
(g ) := (gij ) . The geodesics of F are characterized locally by
dx
d2 xi
+ 2Gi (x, ) = 0
dt2
dt
where
Gi =
∂gjk
1 il
∂gjl
g
2 k −
yj yk .
4
∂x
∂xl
(5)
Definition 2.2. The coefficients of the Riemann curvature Ry = Rki dxi ⊗
given by
2 i
∂Gi
∂ 2 Gi j
∂Gi ∂Gj
j ∂ G
Rik := 2 k −
y
+
2G
−
∂x
∂xj ∂y k
∂y j ∂y k
∂y j ∂y k
i
∂R
∂Ri
∂ 2 Ri
∂ 2 Ri
and Rijk := 31 ∂ykj − ∂yjk , Rji kl := 13 ∂yj ∂ykl − ∂yj ∂ylk .
The Ricci scalar function of F is given by Ric :=
Ricci scalar is the Ricci tensor
1 2
F Ric
.
Ricij :=
2
yi yj
1
i
F 2 Ri .
∂
∂xi
are
(6)
A companion of the
(7)
S. Azami and A. Razavi
160
Definition 2.3. A Finsler metric is said to be an Einstain metric if the Ricci
scalar function is a function of x alone,equivalently Ricij = R(x)gij (see [14, 19]).
Definition 2.4. Let (M, F ) be a Finsler manifold, the Sasakian metric ge of g on
T M0 is defined as
δy i
δy j
ge = gij dxi ⊗ dxj + gij
⊗
(8)
F
F
∂
δ
then ge is a Riemannian metric on T M0 and { δxi , F ∂yi } is a coordinate base on
T M0 , where δxδ i =
δy i = dy i + Gij dxj .
∂
∂xi
i
δ
∂
− Gji ∂y∂ j and {dxi , δy
F } is the dual of { δxi , F ∂y i } where
e on T M0 with respect to the Sasakian
Remark. The Levi-Civita connection ∇
metric ge is locally expressed as follows:
δ
1
∂
e δ δ
∇
= −(Cikj + Rkij ) k + Fikj k
(9)
δxj δxi
2
∂y
δx
∂
δ
e ∂ ∂
= Cikj k − gih (Fjhk − Ghj k )g hk k
∇
i
j
∂y ∂y
∂y
δx
∂
1
δ
∂
e δ
= Fikj k + (Cikj + gih Rhlj g lk ) k
∇
δxj ∂y i
∂y
2
δx
e ∂ δ + Gkij ∂ ,
= ∇
∂y i δxj
∂y k
where
Cikj =
∂Gkj
1 kh ∂gij
1 kh δghi
δghj
δgij
k
k
g
,
F
=
g
(
+
−
),
G
=
,
i
j
i
j
2
∂y h
2
δxj
δxi
δxh
∂y i
Rkij =
δGkj
δ
δ
∂
δ
∂
∂
δGki
−
, [ i , j ] = Rkij k , [ i , j ] = Gkij k .
j
i
δx
δx
δx δx
∂y
δx ∂y
∂y
and
Lemma 2.5. For a Sasakian metric ge and any f : T M → R, there exists a unique
vector field Y ∈ X (T M ) such that
where
and X1i , X2i are C ∞
e = df (X),
e ∀X
e ∈ X (T M )
ge(Y, X)
(10)
e = Xi δ + XiF ∂
X
1
2
δxi
∂y i
function on T M . Here we take Y = 0 if df = 0.
˜ . We call ∇f
˜ the gradient of f and
Denote the vector field Y in (10 ) by ∇f
˜
define the divergence div X as follows:
˜ = tr∇
˜X
˜
div X
Eigenvalues variation of the P -Laplacian
161
Definition 2.6. According to the above definition, the gradient of a function f is
˜ = g ij δf δ + F 2 g ij ∂f ∂
∇f
δxi δxj
∂y i ∂y j
(11)
˜ with respect to the Riemannian metric ge is given
therefore, the norm of ∇f
by
˜ |2 = ge(∇f,
e ∇f
e ) = g ij δf δf + F 2 g ij ∂f ∂f .
|∇f
δxi δxj
∂y i ∂y j
(12)
Definition 2.7. Let M be a compact Finsler manifold. The Laplace operator of f
on T M is defined as follows:
∆f
=
+
−
∂Grj ∂f
∂2f
∂2f
∂2f
∂2f
−
− Grj i r − Gsi s j + Grj Gsi s r )
i
j
i
r
∂x ∂x
∂x ∂y
∂x ∂y
∂y ∂x
∂y ∂y
2
1
δf
δf
∂ f
(13)
g ij F 2 i j + g ij (Cikj + Rkij ) k − F g ij Fikj k
∂y ∂y
2
δx
δx
∂f
δf
F g ij Cikj k − F 2 g ij gih (Gjhl − Fjhl )g lk k .
∂y
δy
g ij (
Definition 2.8. Let M be a compact Finsler manifold. The p-Laplace operator of
f : SM → R, f ∈ W 1,p (SM ) for 1 < p < ∞ is defined as follows:
△p f
=
=
where
e |p−2 ∇f
e )
div(|∇f
e |p−2 ∆f + (p − 2)|∇f
e |p−4 (Hessf )(∇f,
e ∇f
e )
|∇f
(14)
e ∇f
e )(X, Y ) = Y.(X.f ) − (∇
e Y X).f, X, Y ∈ X (SM )
(Hessf )(X, Y ) = ∇(
and in local coordinate, we have:
e kij ∂k f.
(Hessf )(∂i , ∂j ) = ∂i ∂j f − Γ
Note. If f is a function of x alone, or suppose that is the lifting of f : M → R
then
2
∂ f
e kij ∂f
−
Γ
(15)
∆f = g ij
∂xi ∂xj
∂xk
e
e k is christoffel symbol of ∇.
where Γ
ij
2.1. Eigenvalues of the p-Laplacian.
Definition 2.9. Let (M n , F ) be a compact Finsler manifold and f : SM → R. We
say that λ is an eigenvalue of the p-Laplace operator whenever
g
e
∆p f + λ|f |p−2 f = 0
(16)
then f is said to be the eigenfunction associated to λ, or equivalently they satisfy
in
Z
Z
p−2
e
e
e
|f |p−2 f ϕdv
∀ϕ ∈ W01,2 (SM )
(17)
|∇f |
< ∇f, ∇ϕ > dv = λ
SM
SM
S. Azami and A. Razavi
162
where W01,p (SM ) is closure of C0∞ (SM ) in Sobolev W 1,p (SM ).
By substitution ϕ = f in (17) we have:
R
e |p dv
|∇f
λ = RSM
.
|f |p dv
SM
Normalized eigenfunctions are defined as follows:
Z
Z
f |f |p−2 dv = 0,
|f |p dv = 1.
SM
(18)
(19)
SM
Suppose that (M n , Ft ) is a solution of the Ricci flow on the smooth manifold
(M n , F0 ) in the interval [0, T ) and
Z
e (x, y)|p dvt
λ(t) =
|∇f
(20)
SM
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) on M n
Specp (e
g ) = {0 = λ0 (g) ≤ λ1 (g) ≤ λ2 (g) ≤ ... ≤ λk (g) ≤ ...}
is the spectrum of ∆p = ge∆p . In what follows we assume the existence and C 1 differentiability of the elements λ(t) and f (t), under a Ricci flow deformation g(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 for λ(t) under the
Ricci flow. Let (M n , Ft ), t ∈ [0, T ),be a deformation of Finsler metric F0 . Assume
that λ(t) is the eigenvalue of ∆p , f = f (x, y, t) satisfies
∆p f + λ|f |p−2 f = 0
and
R
SM
|f |p dv = 1, using (12), we have:
d e 2
|∇f |
dt
=
+
+
∂ ij δf δf
∂ δf δf
(g ) i j + g ij ( i ) j
∂t
δx δx
∂t δx δx
δf ∂ δf
∂(F 2 ) ij ∂f ∂f
g ij i ( j ) +
g
δx ∂t δx
∂t
∂y i ∂y j
∂
∂f ∂f
∂f ′ ∂f
F 2 (g ij ) i j + 2F 2 g ij i j .
∂t
∂y ∂y
∂y ∂y
(21)
where
∂ ij
∂
(g ) = −g il g jk (glk )
∂t
∂t
(22)
Eigenvalues variation of the P -Laplacian
163
and
∂ δf
(
)
∂t δxi
∂ ∂f
r ∂f
−
G
i
∂t ∂xi
∂y r
∂f ′
∂f ′
∂
∂f
− Gri r − (Gri ) r
i
∂x
∂y
∂t
∂y
δf ′
∂
∂f
− (Gri ) r
i
δx
∂t
∂y
=
=
=
(23)
therefore, a substitution of (22) and (23) in (21), implies that:
Proposition 3.1. Let (M n , Ft )) be a deformation of Finsler manifold (M n , F0 ), then
d e p
|∇f |
dt
∂
p e p−2
δf δf
δf ′ δf
−g il g jk (glk ) i j + 2g ij i j
|∇f |
2
∂t
δx δx
δx δx
∂f δf
∂F ij ∂f ∂f
∂
p e p−2
|
g
2F
−2g ij (Gri ) r j + |∇f
∂t
∂y δx
2
∂t
∂y i ∂y j
∂f ∂f
∂f ′ ∂f
∂
−F 2 g il g jk (glk ) i j + 2F 2 g ij i j .✷
∂t
∂y ∂y
∂y ∂y
=
On the other hand we have
∂
∂
d
(dv) = g ij (gij ) − n (logF ) dv.
dt
∂t
∂t
(24)
Now, we get the following two integrability conditions:
0=
d
dt
Z
|f |p−2 f dv = (p − 1)
SM
Z
|f |p−2 f ′ dv +
SM
Z
|f |p−2 f
SM
d
dv
dt
therefore
(p − 1)
Z
|f |
p−2 ′
f dv = −
SM
Z
|f |
p−2
SM
f
∂
∂
(gij ) − n (logF ) dv
g
∂t
∂t
ij
(25)
and
0=
d
dt
Z
|f |p dv = p
SM
Z
f f ′ |f |p−2 dv +
SM
Z
|f |p
SM
d
dv
dt
which implies
p
Z
′
f f |f |
SM
p−2
∂
ij ∂
(gij ) − n (logF ) dv.
dv = −
|f | g
∂t
∂t
SM
Z
p
(26)
S. Azami and A. Razavi
164
Now, if we suppose that g(t) is a solution of the un-normalized Ricci flow (1) and
(2), then we have:
Z
Z
dλ
d e p
e |p d (dv)
=
( |∇f | )dv +
|∇f
dt
dt
dt
SM
SM
Z
p
δf δf
δf ′ δf
=
−g il g jk (−2Riclk ) i j + 2g ij i j
2 SM
δx δx
δx δx
Z
p
∂f ∂f
r ∂f δf
p−2
ij ∂
e
(G )
|∇f | dv +
2F 2 (−Ric)g ij i j
−2g
∂t i ∂y r δxj
2 SM
∂y ∂y
′
∂f ∂f
2 il jk
2 ij ∂f ∂f
e |p−2 dv
−F g g (−2Riclk ) i j + 2F g
|∇f
∂y ∂y
∂y i ∂y j
Z
e |p g ij (−2Ricij ) − n(−Ric) dv
|∇f
+
Z SM
Z
e ∇f
e )|∇f
e |p−2 dv + p
e ′ , ∇f
e )|∇f
e |p−2 dv
= p
Ric(∇f,
ge(∇f
SM
SM
Z
e |p −2g ij Ricij + nRic dv
|∇f
+
SM
Z
∂f δf e p−2
d
| dv
g ij (Gri ) r j |∇f
−p
dt
∂y
δx
SM
Z
∂f ∂f e p−2
RicF 2 g ij i j |∇f
−p
| dv
∂y ∂y
SM
where
d
r
dt (Gi )
Hence
d
(Gri )
dt
is obtained as follows:
∂gjk
1 rl
∂gjl
r
G = g
2 k −
yj yk ,
4
∂x
∂xl
=
Gri =
∂Gr
∂y i
∂(gab )
1
∂
∂gjl
∂gjk
2
yj yk
− g rα g aβ (gαβ )g lb
−
4
∂t
∂y i
∂xk
∂xl
∂(gab )
1
∂gjl
∂
∂gjk
− g lα g bβ (gαβ )g ra
2
yj yk
−
4
∂t
∂y i
∂xk
∂xl
′
)
∂gjl
∂gjk
1 ra lb ∂(gab
2 k −
yj yk
− g g
4
∂y i
∂x
∂xl
′
′
∂gjl
∂gjk
∂(gab )
1
− g ra g lb
2
−
yj yk
(27)
4
∂y i
∂xk
∂xl
∂(gαβ )
∂ 2 gjl
∂ 2 gjk
1
yj yk
2 i k −
+ g rα g lβ
4
∂t
∂y ∂x
∂y i ∂xl
(
)
′
′
∂ 2 gjl
∂ 2 gjk
1
+ g rl 2 i k −
yj yk
4
∂y ∂x
∂y i ∂xl
′
′
∂(gαβ )
1 rl
∂gil
∂gil
∂gik
∂gik
1
k
y
+
yk
2 k −
2
g
−
+ g rα g lβ
4
∂t
∂x
∂xl
4
∂xk
∂xl
′
′
∂gjl
∂gij
∂(gαβ )
1 rl
∂gjl
∂gij
1
j
y
+
yj .
2 i −
2
g
−
+ g rα g lβ
4
∂t
∂x
∂xl
4
∂xi
∂xl
Eigenvalues variation of the P -Laplacian
165
From the un-normalized Ricci flow, we can then write
d
(Gri )
dt
=
∂(gab )
1 rα aβ
∂gjl
∂gjk
2
yj yk
g g Ricαβ g lb
−
2
∂y i
∂xk
∂xl
∂(gab )
∂gjl
∂gjk
1
2
yj yk
−
+ g lα g bβ Ricαβ g ra
2
∂y i
∂xk
∂xl
∂(Ricab )
∂gjl
∂gjk
1
2
yj yk
−
+ g ra g lb
2
∂y i
∂xk
∂xl
∂(gab )
1
∂Ricjl
∂Ricjk
+ g ra g lb
2
yj yk
−
2
∂y i
∂xk
∂xl
∂ 2 gjl
∂ 2 gjk
1
yj yk
− g rα g lβ Ricαβ 2 i k −
2
∂y ∂x
∂y i ∂xl
2
∂ Ricjl
∂ 2 Ricjk
1
yj yk
− g rl 2 i k −
2
∂y ∂x
∂y i ∂xl
1
∂gil
∂gik
− g rα g lβ Ricαβ 2 k −
yk
2
∂x
∂xl
∂Ricik
∂Ricil
1 rl
−
yk
2
− g
2
∂xk
∂xl
∂gjl
∂gij
1
yj
− g rα g lβ Ricαβ 2 i −
2
∂x
∂xl
1
∂Ricij
∂Ricjl
− g rl 2
yj .
−
2
∂xi
∂xl
Using (26) we obtain
Z
e ′ , ∇f
e )|∇f
e |p−2 dv
p
ge(∇f
=
SM
=
pλ
Z
−λ
(28)
f ′ f |f |p−2 dv
SM
Z
SM
We have thus proved the following proposition:
|f |p −2g ij Ricij + nRic dv.
Proposition 3.2. Let (M n , Ft )) be a solution of the un-normalized Ricci flow on the
smooth Finsler manifold (M n , F0 ). If λ(t) denotes the evolution of an eigenvalue
under the Ricci flow, then
Z
dλ
e ∇f
e )|∇f
e |p−2 dv
Ric(∇f,
= p
dt
SM
Z
e |p ) 2g ij Ricij − nRic dv
+
(λ|f |p − |∇f
(29)
SM
Z
∂
∂f δf e p−2
g ij (Gri ) r j |∇f
−p
| dv
∂t
∂y δx
ZSM
∂f ∂f e p−2
−p
Ric F 2 g ij i j |∇f
| dv
(30)
∂y ∂y
SM
where f is the associated normalized evolving eigenfunction.✷
S. Azami and A. Razavi
166
Note. Let f : SM → R be a lifting of f : M → R. We have:
Z
dλ
e ∇f
e )|∇f
e |p−2 dv
= p
Ric(∇f,
dt
SM
Z
e |p ) −2g ij Ricij + nRic dv
(λ|f |p − |∇f
+
SM
ij
and in this case, if −2g Ricij + nRic is a constant, then
Z
dλ
e ∇f
e )|∇f
e |p−2 dv.
=p
Ric(∇f,
dt
SM
Corollary 3.3. Let (M n , Ft ) be a solution of the un-normalized Ricci flow on the
smooth Riemannian manifold (M n , F0 ), i.e. Ft , F0 are Riemannian metric. If λ(t)
denotes the evolution of an eigenvalue under Ricci flow, then:
Z
Z
dλ
p−2
e |p )dv
e
e
e
R(λ|f |p − |∇f
Ric(∇f, ∇f )|∇f | dv +
= p
dt
SM
SM
Z
∂f δf e p−2
∂
| dv
(31)
−p
g ij (Gir ) r j |∇f
∂t
∂y
δx
SM
Z
∂f ∂f e p−2
p
| dv
RF 2 g ij i j |∇f
−
n SM
∂y ∂y
where R is the scalar curvature of M .
Proof :
If Ft is the Riemannian metric, then
Ric =
1
R,
n
(32)
and
2g ij Ricij − nRic = R,
therefore (31) is obtained by replacing (32) and (33) in (29).
✷
(33)
Corollary 3.4. Let (M 2 , Ft ) be a solution of the un-normalized Ricci flow on the
smooth Riemannian surface (M 2 , F0 ). If λ(t) denotes the evolution of an eigenvalue
under the Ricci flow, then:
Z
Z
p
dλ
p
e |p )dv
e
R(λ|f |p − |∇f
=
R|∇f | dv +
dt
2 SM
SM
Z
∂
∂f δf e p−2
| dv
−p
g ij (Gir ) r j |∇f
∂t
∂y
δx
SM
Z
∂f ∂f e p−2
p
| dv
RF 2 g ij i j |∇f
−
n SM
∂y ∂y
where R is the scalar curvature of M .
Proof : In dimension n = 2, for a Riemannian manifold, we have:
1
Ric = Rg,
2
(34)
Eigenvalues variation of the P -Laplacian
167
hence the corollary is obtained by replacing (34) in (29). ✷
Corollary 3.5. Let (M n , Ft ) be a solution of the un-normalized Ricci flow on the
smooth homogenous Riemannian manifold (M n , F0 ). If λ(t) denotes the evolution
of an eigenvalue under the Ricci flow, then:
dλ
dt
=
p
Z
SM
Z
e ∇f
e )|∇f
e |p−2 dv
Ric(∇f,
∂f δf e p−2
∂
(Gir ) r j |∇f
| dv
∂t
∂y
δx
SM
Z
∂f ∂f e p−2
p
F 2 g ij i j |∇f
| dv
−R
n SM
∂y ∂y
−p
g ij
where R is the scalar curvature of M .
Proof :
Since the evolving metric remains homogenous and a Riemannian homogenous 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 (M n , Ft ) be a solution of the normalized Ricci flow on the
smooth Finsler manifold (M n , F0 ). If λ(t) denotes the evolution of an eigenvalue
under Ricci flow, then:
dλ
dt
=
Z
e ∇f
e )|∇f
e |p−2 dv
Ric(∇f,
(n − p)rλ + p
SM
Z
e |p ) 2g ij Ricij − nRic dv
+
(λ|f |p − |∇f
SM
Z
∂f δf e p−2
∂
g ij (Gsi ) s j |∇f
−p
| dv
∂t
∂y δx
SM
Z
∂f ∂f e p−2
−p
F 2 (Ric − r) g ij i j |∇f
| dv
∂y ∂y
SM
where f is the associated normalized evolving eigenfunction, r =
Ric dv
vol(SM ) .
SM
Proof: In the normalized case, the integrability conditions read as follows
Z
Z
′
p−2
(36)
|f |p 2g ij Ricij − nr − nRic dv.
f f |f | dv =
p
SM
Since
R
(35)
SM
d
(dv) = −2g ij Ricij + nr + nRic dv
dt
(37)
S. Azami and A. Razavi
168
using (24), (27) and the above equation, we can then write
dλ
dt
=
=
Z
Z
d
d
e |p
e |p )dv +
|∇f
|∇f
(dvt )
dt
dt
SM
SM
)
(
Z
′
p
δf
δf δf
ij δf
il jk
e |p−2 dv
|∇f
+ 2g
−g g (−2Riclk + 2rglk )
i
j
i
j
2 SM
δx δx
δx δx
Z
p
2
ij ∂f ∂f
+
2F (−Ric + r)g
2 SM
∂y i ∂y j
∂f ∂f
2 il jk
e |p−2 dv
|∇f
−F g g (−2Riclk + 2rglk )
∂y i ∂y j
)
(
Z
′
∂f
p
ij ∂
2 ij ∂f
r ∂f δf
e |p−2 dv
|∇f
−2g
+ 2F g
(Gi )
+
r
j
i
j
2 SM
∂t
∂y δx
∂y ∂y
Z
n
o
e |p −2g ij Ricij + nr + nRic dv
|∇f
+
(
SM
=
(n − p)rλ + p
+
Z
−p
SM
Z
e |
|∇f
g
SM
ij
p
Z
n
d
dt
SM
−2g
r
e ∇f
e )|∇f
e |p−2 dv + p
Ric(∇f,
ij
(Gi )
o
Ricij + nRic dv
∂f
δf
∂y r δxj
Z
e |
|∇f
p−2
Z
SM
(38)
e ′ , ∇f
e )|∇f
e |p−2 dv
g
e(∇f
dv
p−2
2
ij ∂f ∂f e
|∇f |
dv
F (Ric − r) g
−p
∂y i ∂y j
SM
but
p
Z
SM
e ′ , ∇f
e )|∇f
e |p−2 dv
ge(∇f
=
=
Z
f ′ f |f |p−2 dv
pλ
Z SM
λ
|f |p 2g ij Ricij − nr − nRic dv (39)
SM
∂
′
and ∂t
(Gsi ) is obtained by replacing F ′ and gij
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 (M n , Ft ) be a solution of the normalized Ricci flow on the
smooth Riemannian manifold (M n , F0 ). If λ(t) denotes the evolution of an eigenvalue under Ricci flow, then:
Z
dλ
e ∇f
e )|∇f
e |p−2 dv
= (n − p)rλ + p
Ric(∇f,
dt
SM
Z
Z
∂
∂f δf e p−2
p
p
e
g ij (Gsi ) s j |∇f
R(λ|f | − |∇f | )dv − p
+
| dv (40)
∂t
∂y δx
SM
SM
Z
∂f ∂f e p−2
1
| dv
−p
( R − r)F 2 g ij i j |∇f
n
∂y
∂y
SM
where R is the scalar curvature of M .
Corollary 3.8. Let (M 2 , Ft ) be a solution of the normalized Ricci flow on the
smooth Riemannian surface (M 2 , F0 ). If λ(t) denotes the evolution of an eigenvalue
Eigenvalues variation of the P -Laplacian
169
under Ricci flow, then:
Z
p
dλ
e |p dv
= (2 − p)rλ +
R|∇f
dt
2 SM
Z
Z
∂
∂f δf e p−2
e |p )dv − p
g ij (Gsi ) s j |∇f
R(λ|f |p − |∇f
+
| dv
∂t
∂y δx
SM
SM
Z
∂f ∂f e p−2
R
| dv
−p
( − r)F 2 g ij i j |∇f
∂y ∂y
SM 2
where R is the scalar curvature of M .
Corollary 3.9. Let (M n , Ft ) be a solution of the normalized Ricci flow on the
smooth homogenous Riemannian manifold (M n , F0 ). If λ(t) denotes the evolution
of an eigenvalue under Ricci flow, then:
Z
dλ
e ∇f
e )|∇f
e |p−2 dv
Ric(∇f,
= (n − p)rλ + p
dt
SM
Z
∂f δf e p−2
∂
| dv
−p
g ij (Gsi ) s j |∇f
∂t
∂y
δx
ZSM
1
∂f ∂f e p−2
( R − r)F 2 g ij i j |∇f
−p
| dv
∂y ∂y
SM n
where R is the scalar curvature of M .
4. Examples
In this section, we will find the variational formula for some of Finsler manifolds.
Example 4.1. Let (M n , F0 ) be an Einstein manifold i.e. there exists a constant
a such that Ric(F0 ) = aF02 . 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
where u(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
a
g(t),
Ric(g(t)) = Ric(g0 ) = ag0 =
1 − 2at
a
1
Ric(g0 ) =
,
Ric(gt ) =
1 − 2at
1 − 2at
Ft2 = (1 − 2at)F02
therefore
2g ij Ricij − nRic =
an
1 − 2at
S. Azami and A. Razavi
170
and
Also
e ∇f
e )=
Ric(∇f,
Gi (t)
=
=
a
a
e ∇f
e )=
e |2 .
ge(∇f,
|∇f
1 − 2at
1 − 2at
1 il ∂gjl
∂gjk j k
g {2 k −
}y y
4
∂x
∂xl
∂(g0 )jk j k
∂(g0 )jl
1
−
}y y = Gi (0)
(g0 )il {2
4
∂xk
∂xl
therefore
∂
(Gi ) = 0.
∂t r
Using the un-normalized Ricci flow equation (2) and (29) ,we obtain the following
relation:
Z
Z
a
an
dλ
e |p dv + λ
|f |p
= p
|∇f
dv
dt
1 − 2at
1 − 2at
Z M
Z M
an
∂f ∂f e p−2
a
e |p dv − p
|∇f
F 2 g ij i j |∇f
| dv
−
1
−
2at
1
−
2at
∂y
∂y
M
SM
Z
pa
∂f ∂f e p−2
=
λ−
| dv
F 2 g ij i j |∇f
1 − 2at
∂y
∂y
SM
Now, If we suppose that
R gt = u(t)g0 , u(0) = 1 is a solution of the normalized Ricci
1
flow and r = V ol(SM
) SM Ricdv, then from (3) we have
u′ g0 = −2ag0 + 2rug0 .
It implies that
a
(1 − e2rt ) + e2rt .
r
o
na
(1 − e2rt ) + e2rt
g(t) =
r
u(t) =
So that we have:
and
−1
a
2rt
2rt
Ric(g(t)) = a
g0
(1 − e ) + e
r
−1
a
(1 − e2rt ) + e2rt
Ric(g(t)) = a
r
−1
a
ij
2rt
2rt
2g Ricij − Ricij = an
(1 − e ) + e
,
r
also
Gi (t) = Gi (0),
therefore
∂Gri
= 0.
∂t
Eigenvalues variation of the P -Laplacian
171
Using (35), we obtain the following:
−1
a
dλ
2rt
2rt
= (n − p)rλ + paλ
(1 − e ) + e
dt
r
−1
Z
a
∂f 2 e p−2
− r (y i
) |∇f | dv
− p
a
(1 − e2rt ) + e2rt
r
∂yi
SM
−1
a
2rt
2rt
=
(n − p)r + pa
(1 − e ) + e
λ
r
Z
−1
∂f ∂f e p−2
a
F 2 g ij i j |∇f
−r
(1 − e2rt ) + e2rt
| dv.
− p a
r
∂y ∂y
SM
Remark. Let (M n , F0 ) be a Finsler manifold of dimension n ≥ 3. Suppose that the
flag curvature k = k(x) is isotropic and a function of x ∈ M alone then k = constant
and therefore (M n , 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
3η
Ric = (n − 1){ + σ}F 2
F
where η is a 1-form and σ = σ(x) is scalar function.
Example 4.3. If we suppose that Ft = u(t)F0 , u(0) = 1 is a solution of the Ricci
flow, then:
Ric(Ft ) =
2
0
0
(n − 1){ 3η
(n − 1){ 3η
Ric(F0 )
Ric(Ft )
F0 + σ0 }F0
F0 + σ0 }
=
=
=
Ft2
(u(t))2 F02
(u(t))2 F02
(u(t))2
Now the Ricci flow (1) implies that
−
0
(n − 1){ 3η
F0 + σ0 }
(u(t))2
= −Ric =
∂logF
F′
u′ (t)
= t =
∂t
Ft
u(t)
or equivalently
uu′ = (n − 1){
3η0
+ σ0 }
F0
By integration we have:
u2 (t) = −2(n − 1){
3η0
+ σ0 }t + c
F0
with condition u(0) = 1 we have:
u2 (t) = 1 − 2(n − 1){
therefore
Ft2
=
3η0
+ σ0 }t
F0
3η0
1 − 2(n − 1){
+ σ0 }t F02
F0
(41)
S. Azami and A. Razavi
172
and
e ∇f
e ) = 1 (n − 1)g il g js (3η0 F0 )yl ys + 2σ0 (g0 )ls ∇
e if ∇
e j f.
(42)
Ric(∇f,
2
Also we have
0
(n − 1){ 3η
F0 + σ0 }
.
2g ij Ricij −nRic = (n−1)g ij (3η0 F0 )yi yj + 2σ0 (g0 )ij −n
0
1 − 2(n − 1){ 3η
F0 + σ0 }t
(43)
Note that
∂η ∂F
(44)
gij (t) = (1 − 2(n − 1)σ0 t)(g0 )ij − 6(n − 1)t i j .
∂y ∂y
By replacing (42),(43) and (44) in (29) we obtain the variation of an eigenvalue.
Now, if we suppose that
R Ft = u(t)F0 , u(0) = 1 is a solution of the normalized Ricci
1
flow and r = V ol(SM
) SM Ric dv, then from (3), we have:
r−
0
(n − 1) 3η
F0 + σ0
u2 (t)
= r − Ric =
∂logF
u′ (t)
=
∂t
u(t)
. It implies that
uu′ − u2 r = −(n − 1)
3η0
+ σ0 ,
F0
u(0) = 1
which is an ordinary differential equation and has a solution as follow:
n − 1 3η0
u2 (t) =
+ σ0 (1 − e2rt ) + e2rt
r F0
therefore
n − 1 3η0
Ft2 =
+ σ0 (1 − e2rt ) + e2rt F02
r F0
is
obtained
from
(35).
and dλ
dt
Example 4.4. In this example we determine the behavior of the evolving spectrum
on the Ricci solitons.
Let Ft is a solution of the Ricci flow ∂logF
= −Ric. If ϕ ia a time-independent
∂t
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 ,
∂
∂
∂
logF (x, y) = ϕ∗ logFt (x, y) = logFt (ϕ(x), ϕ∗ (y))
∂t
∂t
∂t
since ϕ is a isometry we have
∂
∂
logF (x, y) = logFt (x, y) = −Ric(x, y) = −Ric(ϕ(x), ϕ∗ (y)) = −Ric(F )
∂t
∂t
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
Ft2 = u(t)ϕ∗t F02 .
(45)
Eigenvalues variation of the P -Laplacian
173
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
g
e
e
∆ p ◦ ϕ∗ = ϕ ∗ ◦ g ∆ p .
Therefore given a diffeomorphism ϕ : M → M we have that
ϕ : (SM, ϕ∗ ge) → (SM, ge)
is an isometry, hence we conclude that (SM, ϕ∗ ge), and (SM, ge) have the same
spectrum
Specp (e
g ) = Specp (ϕ∗ ge)
with eigenfunction fk and ϕ∗ fk respectively. If g(t) is a Ricci soliton on (M n , g0 )
then
1
Specp (e
g0 )
Specp (e
g (t)) =
u(t)
so that λ(t) satisfies
dλ
u′ (t)
1
,
=−
.
u(t) dt
(u(t))2
λ(t) =
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
0
if i = j
(46)
if i 6= j
where it is a solution for the un-normalized Ricci flow and φt (y) is a strictly positive
C ∞ homogeneous function of degree zero and p > 0. We use the formula
(xi ) 2(p+1)
p
if i = j
ij
φt (y)
g =
(47)
0
if i 6= j
for i = 1, ..., n, we have
Gi = −
p + 1 (y i )2
2p xi
and
F 2 Ric(g(t)) = 0
therefore Ric(g(t)) = 0 and the un-normalized Ricci flow equation implies that
∂g(t)
=0
∂t
hence, g(t) = g0 and λ(t) = λ(0).
174
S. Azami and A. Razavi
Example 4.7. Suppose that R2 has the metric
gij (x, y) = φt (y)
4(x1 )2 + 1
−2x1
−2x2
1
(48)
which is a solution for the the un-normalized Ricci flow, where φt (y) is a strictly
positive C ∞ homogeneous function of degree zero. We obtain
G1 = 0, G2 = −(y 1 )2
and
F 2 Ric(g(t)) = 0
therefore Ric(g(t)) = 0 and the un-normalized Ricci flow equation implies that
∂g(t)
=0
∂t
hence, g(t) = g0 and λ(t) = λ(0).
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