Uniqueness of closed self-similar solutions to $\sigma_k^{\alpha}$-curvature flow

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by  Shanze Gao

UNIQUENESS OF CLOSED SELF-SIMILAR SOLUTIONS TO
σα-CURVATURE FLOW
k
7
1
SHANZEGAO,HAIZHONGLI,ANDHUIMA
0
2

n Abstract. Byadapting thetest functions introduced byChoi-Daskaspoulos
a [11] and Brendle-Choi-Daskaspoulos [9] and exploring properties of the k-th
J elementary symmetric functions σk intensively, we show that for any fixed k
4 with1≤k≤n−1,anystrictlyconvexclosedhypersurfaceinRn+1 satisfying
2 σkα = hX,νi, with α ≥ k1, must be a round sphere. In fact, we prove a
uniquenessresultforanystrictlyconvexclosedhypersurfaceinRn+1satisfying
] F+C=hX,νi,whereF isapositivehomogeneoussmoothsymmetricfunction
G oftheprincipalcurvatures andC isaconstant.
D
.
h
t
a 1. Introduction
m
Let X :M →Rn+1 be a smooth embedding of a closed, orientable hypersurface
[
in Rn+1 with n≥2, satisfying

2
v (1.1) σkα =hX,νi
2
where ν is the outward unit normal vector field of M, α> 0, 1 ≤k ≤n and σ is
4 k
the k-th elementary symmetric functions of principal curvatures of M.
6
2 This type of equation is important for the following curvature flow
0
(1.2) X˜ =−σαν.
. t k
1
0 Actually, if X is a solution of (1.1), then
7
1 X˜(x,t)=((kα+1)(T −t))1+1kαX(x)
:
v gives rise to the solution of (1.2) up to a tangential diffeomorphism [22]. So in the
i
X same spirit, we call the solutions of (1.1) self-similar solutions of (1.2).
For k =1, G. Huisken proved the following famous result:
r
a
Theorem 1.1 (Huisken, [20]). If M is a closed hypersurface in Rn+1, with non-
negative mean curvature σ and satisfies the equation
1
σ =hX,νi,
1
then M must be a round sphere.
Fork =n,veryrecently,Choi-Daskalopoulos[11],further,Brendle-Choi-Daskalopoulos
[9] proved the following remarkable result:
2010 Mathematics Subject Classification. Primary53C44;Secondary 53C40.
Key words and phrases. σk curvature,self-similarsolution.
The first and the third authors were supported in part by NSFC grant No. 11271213 and
No.11671223.
Thesecondauthor wassupportedinpartbyNSFCgrantNo.11271214 andNo.11671224.
1
2 SHANZEGAO,HAIZHONGLI,ANDHUIMA
Theorem 1.2 (Choi-Daskalopoulos [11], Brendle-Choi-Daskalopoulos [9]). Let M
be a closed, strictly convex hypersurface in Rn+1 satisfying
σα =hX,νi.
n
If α> 1 , then M must be a round sphere; if α= 1 , then M is an ellipsoid.
n+2 n+2
Remark 1.3. The resultsofconvergenceofσα-curvatureflowcouldimply Theorem
n
1.2. Incaseα= 1,Theorem1.2wascontainedintheresultsofB.Chowin[12]. In
n
casen=2,Theorem1.2wasprovedbyB.Andrewsforα=1in[3],byB.Andrews
and X. Chen for 1 ≤ α ≤ 1 in [6]. In case α = 1 , Theorem 1.2 was proved by
2 n+2
B. Andrews in [2]. The more properties of σα-curvature flow were studied by W.
n
J.Firey [15], B.Chow [12], K.Tso [23], B.Andrews [3], P.-F.Guan andL.Ni [19],
B. Andrews, P.-F. Guan and L. Ni [7], etc.
From Theorem 1.1 and Theorem 1.2, the following natural question arises:
Question. For any fixed k with 1≤k ≤n−1, let M be a closed, strictly convex
hypersurface in Rn+1 satisfying (1.1) with α ≥ 1. Can we conclude that M must
k
be a round sphere?
In this paper, we give an affirmative answer to the above question by proving
the following result:
Theorem 1.4. For any fixed k with 1 ≤ k ≤ n−1, let M be a closed, strictly
convex hypersurface in Rn+1 satisfying
σα =hX,νi
k
with α≥ 1. Then M must be a round sphere.
k
Remark 1.5. Theorem 1.1 implies Theorem 1.4 for the case k = 1 and α = 1.
For α = 1, Theorem 1.4 was contained in the results of B. Chow [12, 13] and B.
k
Andrews [1, 2, 4, 5]. For general k and α, there are some partial results under
certain pinching condition of the principal curvatures of hypersurface, see [22], [8]
and [16].
In fact, we prove the following two theorems:
Theorem A. For any fixed k with 1 ≤ k ≤ n, let M be a closed, strictly convex
hypersurface in Rn+1 satisfying
(1.3) σα+C =hX,νi
k
with constants α and C. If either 1≤k ≤n−1, C ≤0, α≥ 1, or, k =n, C <0,
k
α≥ 1 , then M must be a round sphere.
n+2
Remark 1.6. ChooseC =0,TheoremAreducestoTheorem1.4. Whenk =α=1,
TheoremAimpliestheuniquenessofclosedλ−hypersurfacesintroducedbyCheng-
Wei [10].
Let S (λ) denote the k-th power sum of the principal curvatures λ ,··· ,λ ,
k 1 n
defined by S (λ)= n λk.
k i=1 i
Theorem B. For aPny fixed k with k ≥1, let M be a closed, strictly convex hyper-
surface in Rn+1 satisfying
(1.4) Sα+C =hX,νi
k
with constants α and C. If α≥ 1 and C ≤0, then M must be a round sphere.
k
UNIQUENESS OF CLOSED SELF-SIMILAR SOLUTIONS TO σα-CURVATURE FLOW 3
k
Actually, we consider the following general equation
(1.5) F +C =hX,νi,
where F is a homogeneous smooth symmetric function of the principal curvatures
of degree β and C is a constant, which satisfies the following Condition.
Condition 1.7. Suppose F is a smooth function defined on the positive cone
Γ = {µ ∈ Rn|µ > 0,µ > 0,··· ,µ > 0} of Rn, and satisfies the following
+ 1 2 n
conditions:
i) F is positive and strictly increasing, i.e., F >0 and ∂F >0 for 1≤i≤n.
∂λi
ii) F is homogeneous symmetric function with degree β, i.e., F(tλ)=tβF(λ)
for all t∈R .
+
iii) For any i6=j,
∂F λ − ∂F λ
∂λi i ∂λj j ≥0.
λ −λ
i j
iv) For all (y ,...,y )∈Rn,
1 n
1 ∂logF ∂2logF
(1.6) y2+ y y ≥0.
λ ∂λ i ∂λ ∂λ i j
i i i j
i i,j
X X
Remark 1.8. By using Lemma 3.2, one can see that iii) and iv) in Condition 1.7
are equivalent to the convexity of the function F∗(A) = logF(eA) defined on real
n×n symmetric matrices.
Remark 1.9. We call the inequality (1.6) the key inequality of F in this paper,
which plays an important role in our proof. Its σ version appeared in [18] first,
k
later in [14]. We will give another proof in Lemma 2.5 for σ .
k
Remark 1.10. Lemma 2.6 and Lemma 2.7 say that both σα and Sα with α > 0
k k
satisfy Condition 1.7. In fact, any multiplication combination of such functions
satisfies Condition 1.7, such as σ σ and so on.
2 3
For such general F, we prove
Theorem 1.11. Let M be a closed, strictly convex hypersurface in Rn+1 satisfying
(1.7) F +C =hX,νi,
with constant C. For β >1 and C ≤0, if F satisfies Condition 1.7, then M must
be a round sphere.
In our proof, following the idea of Choi-Daskaspoulos [11] and Brendle-Choi-
Daskaspoulos [9], we consider the quantities
n(β−1)
(1.8) Z = F trb− |X|2,
2β
β−1
(1.9) W˜ = F λ−1 − |X|2,
min 2β
where b = (bij) denotes the inverse of the second fundamental form h = (h )
ij
with respect to an orthonormalframe and λ is the smallest principal curvature
min
of the hypersurface. We find that the techniques in Choi-Daskaspoulos [11] and
Brendle-Choi-Daskaspoulos [9] can be carried out effectively on F which satisfies
Condition 1.7. First we apply the maximum principle for W (see Section 4 for
4 SHANZEGAO,HAIZHONGLI,ANDHUIMA
definition of W) to prove that the maximum point of W˜ is umbilic. Then we use
the strong maximum principle of L = ∂F ∇ ∇ for Z to prove Theorem 1.11. In
∂hij i j
particular, Theorem 1.11 holds for F = σα or F = Sα with α > 1. In Theorem
k k k
1
6.3 and Theorem 6.4, we discuss the cases F = σα with 1 ≤ α ≤ 1 and F = Sk,
k k 2 k
respectively.
The structure of this paper is as follows. In Section 2, we give some properties
of the elementary symmetric functions σ and general F satisfying Condition 1.7
k
andprove that both σα and Sα satisfy the key inequality (Lemma 2.7). In Section
k k
3,wederivesomefundamentalformulasfortheclosedhypersurfaceswhichsatisfies
self-similarequation(1.7) withthe generalhomogeneoussymmetric functionF. In
Section4, wedoanalysisatthe maximumpointofW. InSection5wegiveaproof
of Theorem 1.11. Finally in Section 6, we present the proofs of Theorem A and
Theorem B.
Acknowledgments. TheauthorswouldliketothankProfessorXinanMaforhisnice
lectures on σ -problems delivered in Tsinghua University in January 2016. They
k
also would like to thank Professor S.-T. Yau for his constant encouragement.
2. some properties of elementary symmetric functions and the key
inequality
We first collect some basic notations, definitions and properties of elementary
symmetric functions, which are needed in our investigation of σα self-similar solu-
k
tions and general F self-similar solutions.
Let λ = (λ ,··· ,λ ) denote the principal curvatures of M. Throughout this
1 n
paper, we assume that λ ≤λ ≤···≤λ . Denote
1 2 n
σ (λ)=σ (λ(A))= λ λ ···λ .
k k i1 i2 ik
1≤i1<iX2···<ik≤n
For convenience, we set σ (λ) = 1 and σ (λ) = 0 for k > n or k < 0. Let σ (λ)
0 k k;i
denote the symmetric function σ (λ) with λ = 0 and σ (λ), with i 6= j, denote
k i k;ij
the symmetric function σ (λ) with λ = λ = 0. So ∂σk(λ) = σ , ∂2σk(λ) =
k i j ∂λi k−1;i ∂λi∂λj
σ . Remark that without causing ambiguity we omit λ in the notations of
k−2;ij
σ (λ) for simplicity.
k
Definition 2.1. A hypersurface M is said to be strictly convex if λ ∈Γ = {µ∈
+
Rn|µ >0,µ >0,··· ,µ >0} for any point in M.
1 2 n
The following basic properties related to σ will be used directly.
k
UNIQUENESS OF CLOSED SELF-SIMILAR SOLUTIONS TO σα-CURVATURE FLOW 5
k
Proposition 2.2 (See, for example, [21]). For 0 ≤ k ≤ n and 1 ≤ i ≤ n, the
following equalities hold:
σ =σ +λ σ ,
k+1 k+1;i i k;i
n
λ σ =(k+1)σ ,
i k;i k+1
i=1
X
n
σ =(n−k)σ ,
k;i k
i=1
X
n
λ2σ =σ σ −(k+2)σ .
i k;i 1 k+1 k+2
i=1
X
We now turnto provethe keyinequality for σ . First we showtwo lemmas. Let
k
D(k)(λ)=(d ), i,j =0,··· ,m,denote the followingsymmetric(m+1)×(m+1)-
m ij
matrix
σ σ σ ··· σ
k k;1 k;2 k;m
σ σ σ ··· σ
k;1 k;1 k;12 k;1m
 
σ σ σ ··· σ
k;2 k;21 k;2 k;2m ,
 ... ... ... ... ... 
 
σ σ σ ··· σ 
 k;m k;m1 k;m2 k;m
i.e., d =d and  
ij ji
σ (λ), if i=j =0,
k
σ (λ), if i=0, 1≤j ≤m,
d = k;j
ij
σk;i(λ), if 1≤i=j ≤m,
σ (λ), if 1≤i<j ≤m.
k;ij
Lemma 2.3. If λ ∈ Γ and n ≥ 2, then D(k)(λ) is semi-positive definite for
+ n
1≤k ≤n.
Proof. First, since σ = σ = 0 for 1 ≤ i,p,q ≤ n, it is clear that D(n) is
n;i n;pq n
semi-positive definite.
For 1≤ k ≤ n−1, the statement follows by induction on n. In fact, for n = 2,
thesemi-positive-definitenessisprovedbydirectlycomputation. Now,assumethat
the statement is true for n−1. For λ = (λ ,...,λ ), the assumption implies the
1 n
following matrices are semi-positive definite
σ σ σ ··· σ
k;n k;1n k;2n k;n−1,n
σ σ σ ··· σ
k;1n k;1n k;12 k;1,n−1,n
 
D(k) (λ)= σk;2n σk;21n σk;2n ··· σk;2,n−1,n
n−1;n  ... ... ... ... ... 
 
σ σ σ ··· σ 
 k;n−1,n k;n−1,1n k;n−1,2n k;n−1,n 
 
for 1≤k ≤n−1. And, using
σ =σ +λ σ , σ =σ +λ σ (1≤i≤n−1),
k k;n n k−1;n k,i k;in n k−1;in
we obtain
D(k−1) 0 D(k) η
D(k)(λ)=λ n−1;n + n−1;n ,
n n 0 0 ηT σ
(cid:18) (cid:19) (cid:18) k;n(cid:19)
6 SHANZEGAO,HAIZHONGLI,ANDHUIMA
where ηT =(σ ,σ ,σ ,··· ,σ ). For
k;n k;1n k;2n k;n−1,n
σ σ σ ··· σ σ
k;n k;1n k;2n k;n−1,n k;n
σ σ σ ··· σ , σ
k;1n k;1n k;12n k;1,n−1,n k;1n
 
D(k) η σk;2n σk;21n σk;2n ··· σk;2,n−1,n σk;2n
(cid:18) nη−T1;n σk;n(cid:19)= ... ... ... ... ... ... ,
σ σ σ ··· σ σ 
 k;n−1,n k;n−1,1,n k;n−1,2,n k;n−1,n k;n−1,n
 σ σ σ ··· σ σ 
 k;n k;n1 k;n2 k;n,n−1 k;n 
 
by subtracting the first row from the last row and the first column from the last
D(k) η D(k) 0
column,wefindthat n−1;n iscongruentto n−1;n whichissemi-
ηT σ 0 0
(cid:18) k;n(cid:19) (cid:18) (cid:19)
positivedefinite. SoD(k)(λ)issemi-positivedefinite. Thus,theproofiscompleted.
n
(cid:3)
For λ=(λ ,...,λ )∈Γ , let A(k)(λ)=(a ) denote the following matrix
1 n + ij n×n
1 σ σ σ ··· σ
λ1 k−1;1 k−2;12 k−2;13 k−2;1n
σ 1 σ σ ··· σ
 σk−2;21 λσ2 k−1;2 1kσ−2;23 ··· σk−2;2n 
k−2;31 k−2;32 λ3 k−1;3 k−2;3n ,
 ... ... ... ... ... 
 
 σ σ σ ··· 1 σ 
 k−2;n1 k−2;n2 k−2;n3 λn k−1;n
 
i.e.,
1
σ (λ), for i=j,
aij = λi k−1;i

σ (λ), for i6=j.
 k−2;ij
Lemma 2.4. Let ξT = (σ ,σ ,...,σ ). Then the matrix σ A(k) −ξξT
k−1;1 k−1;2 k−1;n k
is semi-positive definite.
Proof. Denote σ A(k)−ξξT =(w ) . Thus
k ij n×n
σ
k−1;iσ , for i=j,
λ k;i
i
w =
ij  1
λ λ (σkσk;ij −σk;iσk;j), for i6=j.
i j
We divide the proof inthree steps.
Step 1. Since the semi-positive-definiteness is preserved under congruent transfor-
mation, we multiply λ to the i-th row and the i-th column of σ A(k) −ξξT for
i k
1≤i≤n. And, let A˜(k) =(a˜ ) denote the new matrix which is defined by
ij n×n
σ (σ −σ ), for i=j,
k;i k k;i
a˜ =
ij σ σ −σ σ , for i6=j.
( k k;ij k;i k;j
We will discuss A˜(k) instead of σ A(k)−ξξT in the following.
k
Step 2. A˜(k) is semi-positive definite if and only if its principal minors are all
non-negative. Let A˜(k) denote the upper-left m×m sub-matrix of A˜(k). For the
m
symmetry of the elemental functions, it suffices to show detA˜(k) ≥0.
m
UNIQUENESS OF CLOSED SELF-SIMILAR SOLUTIONS TO σα-CURVATURE FLOW 7
k
Step 3. detA˜(k) can be calculated as follows.
m
1 σ σ ··· σ
k;1 k;2 k;m
0 σ σ −σ2 σ σ −σ σ ··· σ σ −σ σ
k k;1 k;1 k k;12 k;1 k;2 k k;1m k;1 k;m
detA˜(k) =det0 σkσk;12−σk;1σk;2 σkσk;2−σk2;2 ··· σkσk;2m−σk;2σk;m
m
... ... ... ... ... 
 
0 σ σ −σ σ σ σ −σ σ ··· σ σ −σ2 
 k k;m1 k;m k;1 k k;m2 k;m k;2 k k;m k;m 
 
1 σ σ ··· σ
k;1 k;2 k;m
σ σ σ σ σ ··· σ σ
k;1 k k;1 k k;12 k k;1m
 
σ σ σ σ σ ··· σ σ
=det k;2 k k;12 k k;2 k k;2m
 ... ... ... ... ... 
 
σk;m σkσk;m1 σkσk;m2 ··· σkσk;m
 σ2 σ σ σ σ ··· σσ
k k k;1 k k;2 k k;m
σ σ σ σ σ σ ··· σ σ
k k;1 k k;1 k k;12 k k;1m
 
=σ−2det σkσk;2 σkσk;12 σkσk;2 ··· σkσk;2m
k  ... ... ... ... ... 
 
σ σ σ σ σ σ ··· σ σ 
 k k;m k k;m1 k k;m2 k k;m
=σm−1detD(k). 
k m
By Lemma 2.3, we know detD(k) ≥0. So, detA˜(k) ≥0 which implies σ A(k)−
m m k
ξξT is semi-positive definite. (cid:3)
Withthehelpoftheproceedingtwolemmas,wefinallyobtainthekeyinequality
for σ . It appeared in [18] first, later in [14]. Here we give another proof.
k
Lemma 2.5. For y =(y ,y ,...,y )∈Rn, the following inequality holds
1 2 n
n n
σ σ σ
k−1;iy2+ k−2;ijy y ≥( k−1;iy )2.
λ σ i σ i j σ i
i k k k
i=1 i6=j i=1
X X X
Proof. By Lemma 2.4, we know
1 1
yT( A(k)− ξξT)y ≥0.
σ σ2
k k
(cid:3)
Now we can show that both σα and Sα with α>0 satisfy Condition 1.7.
k k
Lemma 2.6. For i > j, for F = σα or F = Sα with α > 0, Condition 1.7 iii)
k k
holds, i.e.,
∂F ∂F
λ ≥ λ .
i j
∂λ ∂λ
i j
Proof. For F =Sα, it is clear. For F =σα, we have
k k
∂F ∂F
λ − λ =ασα−1(σ λ −σ λ )=ασα−1σ (λ −λ )≥0.
∂λ i ∂λ j k k−1;i i k−1;j j k k−1;ij i j
i j
(cid:3)
8 SHANZEGAO,HAIZHONGLI,ANDHUIMA
Lemma 2.7. For all (y ,y ,...,y )∈Rn, F = σα or F =Sα with α>0 satisfies
1 2 n k k
Condition 1.7 iv), i.e.,
1 ∂logF ∂2logF
y2+ y y ≥0.
λ ∂λ i ∂λ ∂λ i j
i i i j
i i,j
X X
Proof. For F =σα, it is equivalent to Lemma 2.5.
k
For F =Sα, by the Cauchy-Schwarzinequality, we have
k
λk−1 2 λk λk−2 λk−2
i y ≤ i i y2 = i y2,
S i S S i S i
k k k k
(cid:16)Xi (cid:17) (cid:16)Xi (cid:17)(cid:16)Xi (cid:17) Xi
which leads to the key inequality for Sα. (cid:3)
k
Lemma 2.8. If F satisfies Condition 1.7, λ ∈ Γ and λ < λ ≤ λ ≤ ··· ≤ λ ,
+ 1 2 3 n
then for i>j >1, the following equation holds
∂F (λ −λ )2− ∂F (λ −λ )2
∂λi i 1 ∂λj j 1 >0.
(λ −λ )(λ −λ )(λ −λ )
i 1 j 1 i j
Proof. Forthe caseλ =λ ,itis easyto check. Thenwithoutlossofgenerality,we
i j
assume λ >λ for i>j. Actually, for i>j, by Condition 1.7 i) and iii), we have
i j
∂F λ −λ ∂F λ ∂F ∂F λ −λ
i 1 > i ≥ > j 1.
∂λ λ −λ ∂λ λ ∂λ ∂λ λ −λ
i j 1 i j j j i 1
(cid:3)
3. Fundamental formulas of self-similar solution with general F
LetX :Mn →Rn+1beaclosedconvexhypersurface. Supposethate ,e ,··· ,e
1 2 n
isanorthonormalframeonM. Leth=(h )bethesecondfundamentalformonM
ij
with respect to this given frame. And the principal curvatures are the eigenvalues
of the second fundamental form h.
Let us first consider the following general equation
F +C =hX,νi,
where F = F(λ(h)) is a homogeneous symmetric function of the principal curva-
tures of degree β, C is a constant and ν is the outward normal vector field. And,
let L denote the operator L = ∂F ∇ ∇ . We also suppose F > 0 and ( ∂F ) is
∂hij i j ∂hij
positive definite. Inspired by [22], [11] and [9], we have the following proposition.
The summation convention is used unless otherwise stated.
Proposition 3.1. Given a smooth function F :M →Rn+1 described as above, the
following equations hold:
UNIQUENESS OF CLOSED SELF-SIMILAR SOLUTIONS TO σα-CURVATURE FLOW 9
k
∂F
(1) LF =hX,∇Fi+βF − h h (F +C),
jl li
∂h
ij
∂2F
(2) Lh =h hX,e i+h −Ch h − h h
kl klm m kl km lm ijk stl
∂h ∂h
ij st
∂F
− h h h +(β−1)Fh h ,
mj mi kl km ml
∂h
ij
∂2F
(3) Lbkl =hX,∇bkli−bkl+Cδ +bkpbql h h
kl ijp stq
∂h ∂h
ij st
∂F ∂F
+bkl h h −(β−1)Fδ +2bksbptblq h h ,
mj mi kl sti pqj
∂h ∂h
ij ij
(4) L(Ftrb)=hX,∇(Ftrb)i+(β−1)Ftrb−n(β−1)F2
∂F ∂F
+C(nF −trb h h )+2 ∇ F∇ trb
jl li i j
∂h ∂h
ij ij
∂2F ∂F
+Fbkpbqk h h +2Fbksbptbkq h h ,
ijp stq sti pqj
∂h ∂h ∂h
ij st ij
|X|2 ∂F
(5) L = −βF(F +C).
2 ∂h
ii
i
X
Proof. (1) Differentiating (1.7) gives
(3.1) ∇ F =h hX,e i
j jl l
and
∇ ∇ F =h hX,e i+h −h h hX,νi
i j jli l ij jl il
=h hX,e i+h −h h (F +C).
jli l ij jl il
Then, by ∂F h =βF, we obtain
∂hij ij
∂F
LF =∇ FhX,e i+βF − h h (F +C).
l l jl li
∂h
ij
(2) By Codazzi equation and Ricci identity, we obtain
h =h =h +h R +h R .
klji kjli kjil mj mkli km mjli
Then, using Gauss equation we have
∂F
Lh = (h +h R +h R )
kl kjil mj mkli km mjli
∂h
ij
∂F ∂2F ∂F
=∇( h )− h h + h (h h −h h )
l ijk ijk stl mj ml ki mi kl
∂h ∂h ∂h ∂h
ij ij st ij
∂F
+ h (h h −h h )
km ml ij mi jl
∂h
ij
∂2F ∂F ∂F
=∇∇ F − h h − h h h + h h h
l k ijk stl mj mi kl km ml ij
∂h ∂h ∂h ∂h
ij st ij ij
∂2F
=h hX,e i+h −h h (F +C)− h h
klm m kl km lm ijk stl
∂h ∂h
ij st
10 SHANZEGAO,HAIZHONGLI,ANDHUIMA
∂F
− h h h +βFh h .
mj mi kl km ml
∂h
ij
(3) Since h bml =δ , we have
km kl
(3.2) ∇ bkl =−bkpblq∇ h .
j j pq
And,
∇ ∇ bkl =−∇ (bkpblq∇ h )
i j i j pq
=−bkpbql∇ ∇ h +bksbptblq∇ h ∇ h +bkpblsbqt∇ h ∇ h
i j pq i st j pq i st j pq
=−bkpbql∇ ∇ h +2bksbptblq∇ h ∇ h .
i j pq i st j pq
Then, we obtain
∂F ∂F
Lbkl =−bkpbql ∇ ∇ h +2bksbptblq ∇ h ∇ h
i j pq i st j pq
∂h ∂h
ij ij
∂F ∂F
−bkpbql h h h +2bksbptblq h h
pm mq ij sti pqj
∂h ∂h
ij ij
∂2F
=hX,∇bkli−bkl+(F +C)δ +bkpbql h h
kl ijp stq
∂h ∂h
ij st
∂F ∂F
+bkl h h −βFδ +2bksbptblq h h .
mj mi kl sti pqj
∂h ∂h
ij ij
(4) From (3), we have
∂F ∂2F
∇ ∇ trb=hX,∇trbi−trb+n(F +C)+bkpbqk h h
i j ijp stq
∂h ∂h ∂h
ij ij st
∂F ∂F
+trb h h −nβF +2bksbptbkq h h .
mj mi sti pqj
∂h ∂h
ij ij
Furthermore,
∂F ∂F ∂F
L(Ftrb)=2 ∇ F∇ trb+trb ∇ ∇ F +F ∇ ∇ trb
i j i j i j
∂h ∂h ∂h
ij ij ij
∂F ∂F
=2 ∇ F∇ trb+trbhX,∇Fi+trb h
i j ij
∂h ∂h
ij ij
∂F
−trb h h (F +C)+FhX,∇trbi−Ftrb+nF(F +C)
jl li
∂h
ij
∂2F ∂F ∂F
+Fbkpbqk h h +FtrB h h −nF h
ijp stq mj mi ij
∂h ∂h ∂h ∂h
ij st ij ij
∂F
+2Fbksbptbkq h h
sti pqj
∂h
ij
∂F
=hX,∇(Ftrb)i+(β−1)FtrB−n(β−1)F2+C(nF −trB h h )
jl li
∂h
ij
∂F ∂2F ∂F
+2 ∇ F∇ trb+Fbkpbqk h h +2Fbksbptbkq h h .
i j ijp stq sti pqj
∂h ∂h ∂h ∂h
ij ij st ij