Pure and Applied Mathematics Quarterly
Volume 6, Number 2
(Special Issue: In honor of
Michael Atiyah and Isadore Singer )
331—359, 2010
Hyperbolic Geometric Flow (I): Short-time Existence
and Nonlinear Stability
Wen-Rong Dai, De-Xing Kong, Kefeng Liu
Abstract: In this paper we establish the short-time existence and uniqueness theorem for hyperbolic geometric flow, and prove the nonlinear stability
of hyperbolic geometric flow defined on the Euclidean space with dimension
larger than 4. Wave equations satisfied by the curvatures are derived. The
relations of the hyperbolic geometric flow with the Einstein equations and
the Ricci flow are discussed.
Key words and phrases: hyperbolic geometric flow, quasilinear hyperbolic equation, strict hyperbolicity, symmetric hyperbolic system, nonlinear
stability.
2000 Mathematics Subject Classification: 58J45, 58J47.
1. Introduction
This is the first of a series of papers devoted to the study of hyperbolic geometric flow and its applications to geometry and physics. Hyperbolic geometric
flow was first studied by Kong and Liu in [11]. To introduce such flow we were
partially motivated by the Einstein equations in general relativity and the recent
progress in the Hamilton’s Ricci flow, and by the possibility of applying the powerful theory of hyperbolic partial differential equations to geometry. Hyperbolic
geometric flow is a system of nonlinear evolution partial differential equations of
second order, it is very natural to understand certain wave phenomena in nature
Received June 12, 2007.
332
Wen-Rong Dai, De-Xing Kong, Kefeng Liu
as well as the geometry of manifolds, in particular, it describes the wave character of the metrics and curvatures of manifolds. We will see that the hyperbolic
geometric flow carries many interesting features of both the Ricci flow as well as
the Einstein equations. It has many promising applications to both subjects.
The elliptic and parabolic partial differential equations have been successfully
applied to differential geometry and physics. Typical examples are the Hamilton’s Ricci flow and Schoen-Yau’s solution of the positive mass conjecture. A
natural and important question is if we can apply the well-developed theory of
hyperbolic differential equations to solve problems in differential geometry and
theoretical physics. This series of papers is an attempt to apply the hyperbolic
equation techniques to study some geometrical problems and physical problems.
One has found interesting results in these directions, see for example [16] for the
applications of the hyperbolic geometric flow equations to physics. Our results
already show that the hyperbolic geometric flow is a natural and powerful tool
to study some important problems arising from differential geometry and general
relativity such as singularities, existence and regularity. In this paper we study
the basic properties of the hyperbolic geometric flow such as the short-time existence, nonlinear stability and the wave feature of the curvatures. In the sequel
we will study several fundamental problems, for example, long-time existence,
formation of singularities, as well as the physical and geometrical applications.
Let M be an n-dimensional complete Riemannian manifold with Riemannian
metric gij , the Levi-Civita connection is given by the Christoffel symbols
Γkij
1
= g kl
2
½
∂gjl ∂gil ∂gij
+
−
∂xi
∂xj
∂xl
¾
,
where g ij is the inverse of gij . The Riemannian curvature tensors read
k
=
Rijl
∂Γkjl
∂xi
−
∂Γkil
+ Γkip Γpjl − Γkjp Γpil ,
∂xj
The Ricci tensor is the contraction
Rik = g jl Rijkl
and the scalar curvature is
R = g ij Rij .
p
Rijkl = gkp Rijl
.
Hyperbolic Geometric Flow (I)...
333
The hyperbolic geometric flow under the consideration is the following evolution
equation
∂ 2 gij
= −2Rij
(1.1)
∂t2
for a family of Riemannian metrics gij (t) on M . More general hyperbolic geometric flows were also introduced in [11]. A natural and fundamental problem is the
short-time existence and uniqueness theorem of hyperbolic geometric flow (1.1).
In the present paper, we prove the following short-time existence and uniqueness
theorem, the nonlinear stability theorem for Euclidean space, and derive the corresponding wave equations for the curvatures. These results were announced in
Kong and Liu [11].
0 (x)) be a compact Riemannian manifold. Then there
Theorem 1.1. Let (M , gij
exists a constant h > 0 such that the initial value problem

 ∂ 2 g (x, t) = −2R (x, t),
ij
∂t2 ij
gij (x, 0) = g 0 (x), ∂gij (x, 0) = k 0 (x),
ij
∂t
ij
0 (x) is a symmetric
has a unique smooth solution gij (x, t) on M × [0, h], where kij
tensor on M .
The main difficulty to prove this theorem is that, the hyperbolic geometric flow
(1.1) is a system of nonlinear weakly-hyperbolic partial differential equations of
second order. The degeneracy of the system is caused by the diffeomorphism
group of M which acts as the gauge group of the hyperbolic geometric flow.
Because the hyperbolic geometric flow (1.1) is only weakly hyperbolic, the shorttime existence and uniqueness result on a compact manifold does not come from
the standard PDEs theory directly. In order to prove the above short-time existence and uniqueness theorem, using the gauge fixing idea as in the Ricci flow, we
can derive a system of nonlinear strictly-hyperbolic partial differential equations
of second order, thus Theorem 1.1 comes from the standard PDEs theory. On the
other hand, we can reduce the hyperbolic geometric flow (1.1) to a quasilinear
symmetric hyperbolic system of first order, then using the Friedrich’s theory [5]
of symmetric hyperbolic system (more exactly, the quasilinear version [3]) we can
also prove Theorem 1.1.
Noting an important result on nonlinear wave equations (see [10]), we will
find its other interesting application to geometry, by applying this result to the
334
Wen-Rong Dai, De-Xing Kong, Kefeng Liu
wave equations of curvatures. In the present paper we first use it to prove the
nonlinear stability of the flat solution of the hyperbolic geometric flow defined on
the Euclidean space with dimension larger than 4. More precisely, we have
Theorem 1.2. The flat metric gij = δij of the Euclidean space Rn with n ≥ 5 is
nonlinearly stable.
See Section 4 for the precise definition of nonlinear stability which is very
important in general relativity. The key point of the proof of this theorem is the
global existence of classical solutions of the Cauchy problem for the nonlinear
wave equations.
Similar to Hamilton [6], we derive the corresponding wave equations for the
curvatures, for example, we have
Theorem 1.3. Under the hyperbolic geometric flow (1.1), the curvature tensor
satisfies the evolution equation
∂2
Rijkl = 4Rijkl + 2 (Bijkl − Bijlk − Biljk + Bikjl )
∂t2
−g pq (Rpjkl Rqi + Ripkl Rqj + Rijpl Rqk + Rijkp Rql )
Ã
!
q
∂Γpjl ∂Γqik
∂Γpil ∂Γjk
+2gpq
−
,
∂t ∂t
∂t ∂t
(1.2)
where Bijkl = g pr g qs Rpiqj Rrksl and 4 is the Laplacian with respect to the evolving
metric.
The wave equations for the Ricci and scalar curvatures are stated and proved
in Section 5. This is similar to the Ricci flow equation, except that there are
quadratic lower order terms involving the connection coefficients. It turns out
that there is a very rich theory in nonlinear wave equations to deal with such
terms, see [10].
From the above results we can already see that the hyperbolic geometric flow
has many features of the Ricci flow, therefore many well-developed techniques in
the Ricci flow may be applied to the study of this new kind of flow equations. On
the other hand, the hyperbolic geometric flow can also be viewed as the leading
terms of the vacuum Einstein equations. Since the hyperbolic geometric flow
contains the major terms in the Einstein equations, it not only becomes simpler
Hyperbolic Geometric Flow (I)...
335
and more symmetric, but also possesses rich and beautiful geometric properties.
See Section 6 for more detailed discussions on the relations between the hyperbolic
geometric flow and the Einstein equations, and more generally its relations with
other important problems in general relativity.
The paper is organized as follows. In Section 2, using the gauge fixing idea as
in the Ricci flow, we derive a system of nonlinear strictly-hyperbolic partial differential equations of second order. In Section 3 we reduce the hyperbolic geometric
flow (1.1) to a quasilinear symmetric hyperbolic system of first order, and give
the proof of Theorem 1.1. Section 4 is devoted to the nonlinear stability of the
hyperbolic geometric flow defined on the Euclidean space with dimension larger
than 4. In Section 5, we derive the wave equations satisfied by the curvatures,
and illustrate the wave character of the curvatures. Some discussions are given
in Section 6.
2. Strict hyperbolicity of hyperbolic geometric flow
In this section we consider a modified system of evolution equations of the hyperbolic geometric flow, which is strictly hyperbolic so that we can get a solution
for a short time by solving the corresponding Cauchy problem. The solution of
the system (1.1) then comes from the solution of the modified equations.
Let M be a compact n-dimensional manifold. We consider the hyperbolic
geometric flow (1.1) on M , that is,
∂2
gij (x, t) = −2Rij (x, t).
∂t2
(2.1)
Suppose ĝij (x, t) is a solution of the hyperbolic geometric flow (2.1), and ψt :
M → M is a family of diffeomorphisms of M . Let
gij (x, t) = ψt∗ ĝij (x, t)
be the pull-back metrics. We now want to find the evolution equations for the
metrics gij (x, t). Denote by y(x, t) = ϕt (x) = (y 1 (x, t), y 2 (x, t), · · · , y n (x, t)) in
local coordinates. Then
gij (x, t) =
∂y α ∂y β
ĝαβ (y, t)
∂xi ∂xj
(2.2)
336
and
Wen-Rong Dai, De-Xing Kong, Kefeng Liu
·
¸
∂
∂
∂y α ∂y β
gij (x, t) =
ĝαβ (y, t) i
∂t
∂t
∂x ∂xj
∂y α ∂y β d
∂
=
ĝαβ (y(x, t), t) + ĝαβ (y, t)
i
j
∂x ∂x dt
∂t
µ
∂y α ∂y β
∂xi ∂xj
¶
.
Furthermore, we have
µ 2 α¶ β
∂2
∂y α ∂y β d2 ĝαβ
∂
∂ y
∂y
gij (x, t) =
(y(x, t), t) + i
ĝαβ
2
i
j
2
2
∂t
∂x ∂x dt
∂x
∂t
∂xj
µ 2 β¶
µ α¶ β
∂y α ∂
∂ y
∂
∂y
∂y dĝαβ
(2.3)
+ i
ĝ
+
2
αβ
j
2
i
∂x ∂x
∂t
∂x
∂t
∂xj dt
µ β¶
µ α¶
µ β¶
dĝαβ
∂y α ∂
∂y
∂
∂y
∂
∂y
+2 i
+2 i
ĝαβ .
∂x ∂xj
∂t
dt
∂x
∂t ∂xj
∂t
On the other hand,
∂ĝαβ ∂y γ
∂ĝαβ
dĝαβ
(y(x, t), t) =
+
,
γ
dt
∂y ∂t
∂t
d2 ĝαβ
∂ 2 ĝαβ ∂y γ ∂y λ
∂ 2 ĝαβ ∂y γ
∂ 2 ĝαβ
∂ĝαβ ∂ 2 y γ
(y(x,
t),
t)
=
+
2
+
+
dt2
∂y γ ∂t ∂t
∂t2
∂y γ ∂t2
∂y γ ∂y λ ∂t ∂t
and
∂ 2 ĝαβ
(y, t) = −2R̂αβ (y, t).
∂t2
It follows from (2.3) that
∂ 2 ĝαβ ∂y α ∂y β ∂y γ ∂y λ
∂ 2 gij
∂y α ∂y β
(x,
t)
=
−2
R̂
(y,
t)
+
αβ
∂t2
∂xi ∂xj
∂y γ ∂y λ ∂xi ∂xj ∂t ∂t
µ
¶
µ
¶
∂ 2 ĝαβ ∂y α ∂y β ∂y γ
∂
∂y β ∂ 2 y α
∂
∂y β ∂ 2 y α
+2 γ
+ i ĝαβ j
+ j ĝαβ i
∂y ∂t ∂xi ∂xj ∂t
∂x
∂x ∂t2
∂x
∂x ∂t2
·
µ
¶
µ
¶¸
∂ĝαβ ∂y α ∂y β
∂
∂y β
∂
∂y β
∂ 2yγ
+
−
ĝ
−
ĝ
βγ
βγ
∂y γ ∂xi ∂xj
∂xi ∂xj
∂xj ∂xi
∂t2
µ α¶ β µ
¶
∂ĝαβ
∂ĝαβ ∂y γ
∂
∂y
∂y
+2 i
+
∂x
∂t
∂xj
∂t
∂y γ ∂t
µ
¶
µ
¶
∂ĝαβ ∂y γ
∂ĝαβ
∂y β
∂y 2 ∂
+2 i j
+
∂x ∂x
∂t
∂y γ ∂t
∂t
µ α¶
µ β¶
∂
∂y
∂
∂y
+2ĝαβ i
.
j
∂x
∂t ∂x
∂t
(2.4)
Hyperbolic Geometric Flow (I)...
337
Let us choose the normal coordinates {xi } around a fixed point p ∈ M such
∂gij
that
= 0 at p. We next prove that, at p ∈ M ,
∂xk
µ β
¶
µ β
¶
∂ĝαβ ∂y α ∂y β
∂y
∂y
∂
∂
− i
ĝβγ − j
ĝβγ = 0, ∀ i, j, γ = 1, · · · , n.
∂y γ ∂xi ∂xj
∂x ∂xj
∂x
∂xi
(2.5)
The left hand side of (2.5) is
µ β
¶
µ β
¶
∂ĝαβ ∂y α ∂y β
∂
∂y
∂
∂y
− i
ĝβγ − j
ĝαβ
∂y γ ∂xi ∂xj
∂x ∂xj
∂x
∂xi
µ
¶
µ
¶
µ
¶
∂xm ∂xn ∂y α ∂y β
∂
∂xm
∂
∂xm
∂
− i gmj γ − j gmi γ
= γ gmn α
∂y
∂y ∂y β ∂xi ∂xj
∂x
∂y
∂x
∂y
·
µ m¶ n α β
µ n¶ α β¸
m
∂
∂x
∂x ∂y ∂y
∂x ∂
∂x
∂y ∂y
= gmn
+
γ
α
i
j
α
γ
β
β
∂y
∂y
∂y ∂y
∂xi ∂xj
∂y ∂x ∂x
∂y
µ m¶
µ m¶
∂
∂x
∂
∂x
−gmj i
− gmi j
γ
∂x
∂y
∂x
∂y γ
µ m¶ α
µ m¶ β
∂x
∂y
∂
∂x
∂y
∂
+ gmi γ
= gmj γ
α
i
β
∂y
∂y
∂x
∂y
∂xj
∂y
µ m¶
µ m¶
∂
∂x
∂
∂x
−gmj i
− gmi j
γ
∂x
∂y
∂x
∂y γ
µ m¶ α
µ m¶ β
∂
∂x
∂y
∂
∂x
∂y
= gmj α
+ gmi β
γ
i
γ
∂y
∂y
∂x
∂y
∂xj
∂y
µ m¶
µ m¶
∂x
∂
∂x
∂
− gmi j
−gmj i
∂x
∂y γ
∂x
∂y γ
= 0.
So (2.5) holds.
By (2.4) and (2.5), we have
∂ 2 gij
∂
(x, t) = −2Rij (x, t) + i
∂t2
∂x
µ
∂xm ∂ 2 y α
gmj α
∂y ∂t2
¶
∂
+ j
∂x
µ
∂y α ∂ 2 y α
gmi m
∂x ∂t2
∂ 2 ĝαβ ∂y α ∂y β ∂y γ ∂y λ
∂ĝαβ ∂y γ ∂y α ∂y β
+
2
∂y γ ∂t ∂t ∂xi ∂xj
∂y γ ∂y λ ∂xi ∂xj ∂t ∂t
µ α¶ β µ
¶
∂b
gαβ
∂b
gαβ ∂y γ
∂y
∂y
∂
+2 i
+
∂x
∂t
∂xj
∂t
∂y γ ∂t
µ ⶵ
¶
∂b
gαβ
∂b
gαβ ∂y γ
∂y α ∂
∂y
+2 i
+
∂x ∂xj
∂t
∂t
∂y γ ∂t
µ α¶
µ β¶
∂
∂y
∂
∂y
+2b
gαβ i
.
∂x
∂t ∂xj
∂t
¶
+
(2.6)
338
Wen-Rong Dai, De-Xing Kong, Kefeng Liu
We define y(x, t) = ϕt (x) by the following initial value problem
 2 α
◦
∂ y
∂y α jl k

k ),


=
g
(Γ
−
Γ
jl
jl
∂t2
∂xk


 y α (x, 0) = xα , ∂ y α (x, 0) = y1α (x)
∂t
and define the vector filed
µ
◦ ¶
jl
k
Vi = gik g
Γjl − Γkjl ,
(2.7)
◦
where Γkjl and Γkjl are the connection coefficients corresponding to the metrics
gij (x, t) and gij (x, 0), respectively, y1α (x) (α = 1, 2, · · · , n) are arbitrary C ∞
smooth functions on the manifold M . We get the following evolution equation
for the pull-back metric
 2
∂ 2 gbαβ ∂y α ∂y β ∂y γ ∂y λ
∂ gij


(x,
t)
=
−2R
(x,
t)
+
∇
V
+
∇
V
+
ij
i j
j i


∂t2
∂y γ ∂y λ ∂xi ∂xj ∂t ∂t



µ α¶ β µ
¶



∂ 2 gbαβ ∂y α ∂y β ∂y γ
∂b
gαβ
∂b
gαβ ∂y γ
∂
∂y
∂y


+2 γ
+2 i
+


∂y ∂t ∂xi ∂xj ∂t
∂x
∂t
∂xj
∂t
∂y γ ∂t



µ β¶ α µ
¶



∂b
gαβ
∂b
gαβ ∂y γ
∂y
∂y
∂

+
+2 j
∂x
∂t
∂xi
∂t
∂y γ ∂t

µ
¶
µ
¶



∂
∂y α
∂
∂y β


+2b
g
αβ


∂xi
∂t ∂xj
∂t






, −2Rij (x, t) + ∇i Vj + ∇j Vi + F (Dy, Dt Dx y),






∂
 g (x, 0) = g 0 (x),
0 (x),
gij (x, 0) = kij
ij
ij
∂t
(2.8)
where
µ α
¶
µ 2 α¶
∂y ∂y α
∂ y
Dy =
,
, Dt Dx y =
(α, i = 1, 2, · · · , n).
∂t ∂xi
∂xi ∂t
Let
µ
¶
∂y α ∂y α ∂ 2 y α
,
,
(α, i = 1, 2, · · · , n).
∂t ∂xi ∂xi ∂t
b = F (Dy, Dt Dx y) in (2.8) is smooth and F (λ)
b =
The nonlinear term F = F (λ)
b 2 ) holds.
O(|λ|
b=
λ
Since
Γkji =
∂xk ∂ 2 y α
∂y α ∂y β ∂xk b γ
Γαβ + α j i ,
j
i
γ
∂x ∂x ∂y
∂y ∂x ∂x
Hyperbolic Geometric Flow (I)...
the initial value problem (2.7) can be written as
 2 α
µ 2 α
¶
◦
α
β ∂y γ
∂
y
∂
y
∂y
∂y

jl
k
α
b

=g
− Γjl k + Γβγ j
,

∂t2
∂x ∂xi
∂xj ∂xl
∂x

∂ α

 y α (x, 0) = xα ,
y (x, 0) = y1α (x).
∂t
339
(2.9)
At the same time, in the normal coordinates {xi },
½
¾
½ µ
¶¾
∂
∂gil ∂gij
∂
kl ∂gkl
kl ∂gjl
g
− k g
+
−
−2Rij (x, t) + ∇i Vj + ∇j Vi =
∂xi
∂xj
∂xi
∂xj
∂x
∂xl
¶¾
µ
½
∂
1 kl ∂gpl ∂gql ∂gpq
+gjk g pq i
g
+
−
∂x 2
∂xq
∂xp
∂xl
½
µ
¶¾
∂
1 kl ∂gpl ∂gql ∂gpq
+gik g pq j
g
+ (lower order terms)
−
∂x
2
∂xq ∂xp
∂xl
½ 2
¾
∂ 2 gjl
∂ 2 gij
∂ gkl
∂ 2 gil
= g kl
−
−
+
∂xi ∂xj
∂xi ∂xk
∂xj ∂xk
∂xk ∂xl
½ 2
¾
∂ gpj
∂ 2 gqj
∂ 2 gpq
1
+ g pq
+
−
2
∂xi ∂xq
∂xi ∂xq
∂xi ∂xj
½ 2
¾
∂ gpi
∂ 2 gqi
∂ 2 gpq
1
+
−
+ (lower order terms)
+ g pq
2
∂xi ∂xq
∂xj ∂xp ∂xi ∂xj
= g kl
∂ 2 gij
+ (lower order terms).
∂xk ∂xl
Thereby, the initial value problem (2.8) can be written as
 2
2
∂ gij

kl ∂ gij + F (Dy, D D y) + G(g, D g),


(x,
t)
=
g
t x
x
∂t2
∂xk ∂xl

∂

0 (x),
0 (x),
 gij (x, 0) = gij
gij (x, 0) = kij
∂t
µ
¶
∂gij
where g = (gij ), Dx g =
(i, j, k = 1, 2, · · · , n). Let
∂xk
µ
¶
∂gij
µ
b = gij , k
(i, j, k = 1, 2, · · · , n).
∂x
(2.10)
The nonlinear term G = G(b
µ) = G(g, Dx g) in (2.10) is smooth and quadratic
with respect to Dx g.
We observe that both (2.9) and (2.10) are clearly strictly hyperbolic systems.
Since the equations (2.9) and (2.10) are strictly hyperbolic and the manifold M
is compact, it follows from the standard theory of hyperbolic equations (see [8],
Wen-Rong Dai, De-Xing Kong, Kefeng Liu
340
[9], [10]) that the system united by (2.9) and (2.10) has a unique smooth solution
for a short time. Thus, we have proved Theorem 1.1.
3. Symmetrization of hyperbolic geometric flow — second proof
of Theorem 1.1
In this section we reduce the hyperbolic geometric flow (2.1) to a symmetric
hyperbolic system. Then we use the theory of symmetric hyperbolic system to
give another proof of Theorem 1.1.
Let M be a compact n-dimensional manifold and gij (x, t) is a hyperbolic geometric flow on M . We denote the corresponding connection coefficients, the
Riemannian curvature tensor and the Ricci curvature tensor by Γkij , Rijkl and
Rik , respectively.
We consider the space-time R × M with the Lorentzian metric
ds2 = −dt2 + gij (x, t)dxi dxj
(3.1)
and denote the corresponding connection coefficients, the Riemannian curvature
bγ , R
bαβγλ and R
bαβ , respectively. Here
tensor and the Ricci curvature tensor by Γ
αβ
and hereafter, the Greek indices run from 0 to n, Latin indices from 1 to n. The
summation convention is employed. We also denote x0 = t.
By direct computations,
b k = Γk , Γ
b 0 = 1 ∂gij , Γ
bk = Γ
b k = 1 g ki ∂gil , Γ
b 0 = 0, Γ
b i = 0, Γ
b 0 = 0,
Γ
00
00
ij
ij
ij
0i
0l
l0
2 ∂t
2
∂t
bk =
R
ijl
∂Γkjl
∂xi
−
∂Γkil b k b α b k b α
k + 1 g km ∂gmi ∂gjl − 1 g km ∂gmj ∂gil ,
+ Γiα Γjl − Γjα Γil = Rijl
j
∂x
4
∂t ∂t
4
∂t ∂t
li
2
bl = 1 ∂g ∂gik + 1 g li ∂ gik + 1 g lm g pn ∂gmp ∂gnk .
R
0k0
2 ∂t ∂t
2
∂t2
4
∂t ∂t
(3.2)
Hyperbolic Geometric Flow (I)...
341
Then,
bik = g αβ R
biαkβ = g αβ gkl R
bl = −gkl R
bl + g pq gkl R
bl
R
i00
ipq
iαβ
·
¸
2
lp
1 ∂g ∂gip 1 lp ∂ gip 1 lm pn ∂gmp ∂gni
= −gkl −
− g
− g g
2 ∂t ∂t
2
∂t2
4
∂t ∂t
·
¸
1 lm ∂gmp ∂giq
1 lm ∂gmi ∂gpq
pq
l
− g
+g gkl Ripq + g
4
∂t ∂t
4
∂t ∂t
1 ∂ 2 gik
1 ∂g lp ∂gip 1 pn ∂gkp ∂gni
gkl
+ g
+
R
+
ik
2 ∂t2
2
∂t ∂t
4
∂t ∂t
1 pq ∂gkp ∂giq
1 pq ∂gki ∂gpq
− g
+ g
4
∂t ∂t
4
∂t ∂t
∂gip ∂gkq
1 ∂ 2 gik
1 pq ∂gik ∂gpq
1
=
+ Rik + g
− g pq
.
2
2 ∂t
4
∂t ∂t
2
∂t ∂t
=
(3.3)
A direct computation gives (see, e.g., Fock [4], p.423; Fisher and Marsden [3],
p.22)
Ã
!
µ
¶
k
k
bα
bα
∂
Γ
1
∂Γ
∂Γ
1
∂
Γ
(h)
(h)
b +
bij = R
b +
gik j + gjk i ,
R
giα j + gjα i = R
ij
ij
2
∂x
∂x
2
∂x
∂x
where
4 i
b 0 = 1 g kl ∂gkl , Γ
b 0 = g kl Γ
b α , i.e., Γ
b i = g kl Γi =
b α = g βγ Γ
b i = g kl Γ
Γ,
Γ
kl
kl
βγ
kl
2
∂t
µ
¶
∂gαβ
1 αβ ∂ 2 gij
(h)
b
b
Rij = − g
+ Hij gαβ ,
2
∂xα ∂xβ
∂xλ
µ
¶
∂gαβ
1 kl ∂ 2 gij
1 ∂ 2 gij
b
− g
=
+ Hij gαβ ,
2 ∂t2
2 ∂xk ∂xl
∂xλ
and
µ
¶
∂gαβ
b
be Γ
bσ
Hij gαβ ,
= g αβ geσ Γ
iβ jα
∂xλ
µ
¶
1 ∂gij b α
λ αγ βg ∂gγg
λ αγ βg ∂gγg
b
b
Γ + gjλ Γαβ g g
+ giλ Γαβ g g
.
+
2 ∂xα
∂xi
∂xj
Similar to (3.3), we have
µ
¶
1 kl ∂ 2 gij
∂gkl ∂gkl
1 ∂ 2 gij
(h)
b
− g
,
Rij =
+ Hij gkl ,
2 ∂t2
2 ∂xk ∂xl
∂xp ∂t
Wen-Rong Dai, De-Xing Kong, Kefeng Liu
342
and
µ
¶
µ
¶
∂gαβ
∂gkl ∂gkl ∆ b
Hij gkl ,
,
= Hij gαβ ,
∂xp ∂t
∂xλ
p q
kl
b ki0 Γ
b0 Γ
b0
b lj0 + (−1)g kl Γ
= −gkl Γ
ik jl + g gpq Γik Γjl
µ
¶
1 ∂gij 1 kl ∂gkl ∂gij k pq
k ∂gpq pr qs
k pr qs ∂gpq
+
g
+
Γ g + gjk Γrs
g g + gik Γrs g g
2
∂t 2
∂t
∂xk pq
∂xi
∂xj
∂gip 1 lq ∂gjq
1
1 ∂gik 1 ∂gjl
= −gkl g kp
g
− g kl
+ g kl gpq Γpik Γqjl
2
∂t 2
∂t
2 ∂t 2 ∂t
1 ∂gij k pq
1 kl ∂gkl ∂gij
+
Γ g
+ g
4
∂t ∂t
2 ∂xk pq
µ
¶
1
k pr qs ∂gpq
k pr qs ∂gpq
+
gik Γrs g g
+ gjk Γrs g g
2
∂xj
∂xi
∂gip ∂gjq
1
1 ∂gkl ∂gij
= − g pq
+ g kl
2
∂t ∂t
4
∂t ∂t
1 ∂gij k pq
p q
kl
+g gpq Γik Γjl +
Γ g
2 ∂xk pq
µ
¶
1
k pr qs ∂gpq
k pr qs ∂gpq
+
gik Γrs g g
+ gjk Γrs g g
.
2
∂xj
∂xi
It follows from (3.3) that
µ
¶
1 ∂ 2 gij
∂Γk
1 ∂ 2 gij
1 kl ∂ 2 gij
1
∂Γk
gik j + gjk i
+ Rij =
− g
+
2 ∂t2
2 ∂t2
2 ∂xk ∂xl 2
∂x
∂x
¶
µ
∂gip ∂gjq
∂gij ∂gpq
1
1
∂gkl ∂gkl
,
+ g pq
− g pq
,
+Hij gkl ,
p
∂x
∂t
2
∂t ∂t
4
∂t ∂t
i.e.,
µ
¶
2
∂ 2 gij
∂ 2 gij
∂Γk
∂Γk
kl ∂ gij
+ 2Rij =
−g
+ gik j + gjk i
∂t2
∂t2
∂x
∂x
∂xk ∂xl
µ
¶
∂gip ∂gjq
∂gij ∂gpq
∂gkl ∂gkl
1
+2Hij gkl ,
,
+ g pq
− g pq
p
∂x
∂t
∂t ∂t
2
∂t ∂t
¶
µ
2
2
k
k
∂ gij
∂ gij
∂Γ
∂Γ
− g kl k l + gik j + gjk i
=
2
∂t
∂x
∂x
∂x ∂x
∂gij k pq
+2g kl gpq Γpik Γqjl +
Γ g
∂xk pq
¶
µ
k pr qs ∂gpq
k pr qs ∂gpq
+ gjk Γrs g g
.
+ gik Γrs g g
(3.4)
∂xj
∂xi
Hyperbolic Geometric Flow (I)...
343
Similar to the harmonic coordinates in the space-time (see [3]), here we make
use of a new kind of coordinates on the manifold M defined by
∆
Γi = g kl Γikl = 0.
(3.5)
Such new coordinates are called elliptic coordinates on M .
Lemma 3.1. Let gij be a C ∞ Riemannian metric on the manifold M . There
is a C ∞ local coordinates transformation φ : M −→ M , x −→ x around a
fixed point p ∈ M such that the transformed metric g ij is a C ∞ Riemannian
k
metric with Γ (x) = 0 for all x in a neighborhood around p ∈ M and any
k ∈ {1, 2, · · · , n}.
Proof. Consider the elliptic equation for the scalar ψ,
4
4ψ = −g kl
∂2ψ
∂ψ
+ g kl Γjkl
= 0.
∂xk ∂xl
∂xj
The coefficients are C ∞ . Let xi (x) be a solution with the condition xi (p) =
∂xi
(p) = δji . Then xi is a C ∞ local coordinates transformation φ :
xi (p),
∂xj
M −→ M around p and the transformed metric g ij is a C ∞ Riemannian metric.
Now the equation 4ψ = 0 is a tensorial (scalar) equation. In the barred
coordinates, it becomes
0 = 4xk = −g ij
k
∂ 2 xk
j ∂x
k
+Γ
=Γ .
∂xi ∂xj
∂xj
Therefore, g ij satisfies the elliptic condition (3.5). The proof of Lemma 3.1 is
complete. ¶
By Lemma 3.1, we can choose the elliptic coordinates around a fixed point
p ∈ M and for a fixed time t ∈ R+ . After throwing off the bar sign, the geometric
hyperbolic flow (3.1) can be written as
µ
¶
2
∂ 2 gij
∂g
kl
kl ∂ gij
e ij gkl ,
=g
,
(3.6)
+H
∂t2
∂xp
∂xk ∂xl
where
µ
¶
µ
¶
∂gkl
p q
kl
k pr qs ∂gpq
k pr qs ∂gpq
e
Hij gkl ,
= −2g gpq Γik Γjl − gik Γrs g g
+ gjk Γrs g g
∂xp
∂xj
∂xi
(3.7)
344
Wen-Rong Dai, De-Xing Kong, Kefeng Liu
∂gkl
and rational with respect to
∂xp
gkl with non-zero denominator det(g) =
6 0. By introducing the new unknowns
∂gij
∂gij
, gij,k =
gij , hij =
, the system (3.6) can be reduced to a system of
∂t
∂xk
partial differential equations of first order. We now consider such a quasi-linear
1
(symmetric hyperbolic) system with n(n + 1)(n + 2) PDEs of first order
2

∂gij


= hij ,


∂t



∂gij,k
∂hij
(3.8)
g kl
= g kl k ,

∂t
∂x





e ij .
 ∂hij = g kl ∂gij,k + H
∂t
∂xl
are homogenous quadratic with respect to
In the C 2 class, the system (3.8) is equivalent to (3.6).
1
n(n + 1)(n + 2)-dimensional unknown vector
2
function. The coefficient matrices A0 , Aj , B are given by


I 0
0 ··· 0 0
 11 12

 0 g I g I · · · g 1n I 0 


 0 g 21 I g 22 I · · · g 2n I 0 


A0 (u) = A0 (gij , gij,k , hij ) =  .
,
.
. ···



 n1 n2

nn
0
g
I
g
I
·
·
·
g
I
0


0 0
0 ··· 0 I


0 0
0 ··· 0
0


0 · · · 0 g j1 I 
 0 0


 0 0
0 · · · 0 g j2 I 
j
j
,

A (u) = A (gkl , gkl,p , hkl ) = 


· · · · · · · · ·


jn
0 ··· 0 g I
 0 0
0 g 1j I g 2j I · · · g nj I 0
µ
¶ µ
¶
µ
¶
1
1
1
where 0 is the
n(n + 1) ×
n(n + 1) zero matrix, I is the
n(n + 1) ×
2
2
µ
¶ 2
1
n(n + 1) identity matrix,
2


hij


B(u) = B(gij , gij,p , hij ) =  0  ,
e ij
H
Let u = (gij , gij,k , hij )T be the
Hyperbolic Geometric Flow (I)...
345
1 2
n (n + 1)-dimensional zero vector.
2
We observe that the symmetric hyperbolic system
in which 0 is the
A0 (u)
∂u
∂u
= Aj (u) j + B(u)
∂t
∂x
(3.9)
is nothing but the system (3.8). So far, we have reduced the equation of the
hyperbolic geometric flow (3.1) to the symmetric hyperbolic system (3.9), which
are equivalent to each other in the C 2 class. Then, by the theory of the symmetric
hyperbolic system, the smooth solution to (3.1) exists uniquely for a short time
(see [3]). Thus, the proof of Theorem 1.1 is completed.
Remark 3.1. The elliptic coordinates can also be used to prove the short-time
existence for the Ricci flow.
More generally, motivated by general Einstein equations and the rich theory
of hyperbolic equations, we may also consider the following field equations with
the energy-momentum tensor Tij under certain conditions:
µ
¶
∂ 2 gij
∂g
αij
+ 2Rij + Fij g,
= κTij ,
(3.10)
∂t2
∂t
where αij are certain smooth functions on M which may also depend on t, Fij
are some given smooth functions of the Riemannian metric g and its first order
derivative with respect to t, and κ is a parameter. Similar results can be obtained.
Remark 3.2. For noncompact manifolds, if the initial metric, velocity and curvature satisfy some bounded conditions, then a theory similar to Shi’s short-time
existence result can be developed. That is to say, under suitable assumptions,
Shi’s short-time existence result for the ricci flow can be extended to the hyperbolic geometric flow.
4. Nonlinear stability for hyperbolic geometric flow
In this section we investigate the nonlinear stability of the hyperbolic geometric
flow defined on the Euclidean space with the dimension larger than 4.
We now state the definition of nonlinear stability of the hyperbolic geometric
flow (1.1). Let M be an n-dimensional complete Riemannian manifold. Given
346
Wen-Rong Dai, De-Xing Kong, Kefeng Liu
0 and g 1 on M , we consider the following initial value probsymmetric tensors gij
ij
lem
 2

 ∂ gij (x, t) = −2Rij (x, t),
∂t2
(4.1)

gij (x, 0) = g ij (x) + εg 0 (x), ∂gij (x, 0) = εg 1 (x),
ij
ij
∂t
where ε > 0 is a small parameter.
Definition 4.1. The Ricci flat Riemannian metric g ij (x) possesses the (locally)
0 , g 1 ), if there exists a positive constant ε =
nonlinear stability with respect to (gij
0
ij
0
1
ε0 (gij , gij ) such that, for any ε ∈ (0, ε0 ], the initial value problem (4.1) has a
unique (local) smooth solution gij (x, t);
g ij (x) is said to be (locally) nonlinearly stable, if it possesses the (locally) non0 (x) and g 1 (x) with
linear stability with respect to arbitrary symmetric tensors gij
ij
compact support.
In what follows, we consider the nonlinear stability of the flat metric of the
Euclidean space Rn with the dimension n ≥ 5. We have
Theorem 4.1. The flat metric gij = δij of the Euclidean space Rn with n ≥ 5 is
nonlinearly stable.
Remark 4.1. Theorem 4.1 gives the nonlinear stability of the hyperbolic geometric flow on the Euclidean space with dimension larger than 4. The situation for
the 3-, 4-dimensional Euclidean spaces is very different, and will be studied in
the sequel by using null conditions. This is similar to the proofs of the Poincaré
conjecture in topology: the proofs for the three and four dimensional case and
n ≥ 5 dimensional case are very different (see, for example, [6] and [1]). This
motivates us to understand the possibility of using hyperbolic geometric flow to
understand Poincaré conjecture.
Proof of Theorem 4.1.
Define a 2-tensor h in the following way
gij (x, t) = δij + hij (x, t).
Let δ ij be the inverse of δij . Then, for small h
4
H ij = g ij − δ ij = −hij + Oij (h2 ),
where hij = δ ik δ jl hkl and Oij (h2 ) vanishes to second order at h = 0.
Hyperbolic Geometric Flow (I)...
347
As in Section 3, we choose the above elliptic coordinates {xi } around the origin
in Rn . Then the initial value problem (4.1) can be written as
µ
¶
 2
¡ kl
¢ ∂ 2 hij
∂
∂hkl (x, t)

kl

hij (x, t) = δ + H
+ H̃ij δkl + hkl (x, t),
,
∂t2
∂xp
∂xk ∂xl

h (x, 0) = εg 0 (x), ∂ h (x, 0) = εg 1 (x),
ij
ij
ij
ij
∂t
where
µ
¶
∂gkl (x, t)
H̃ij
= H̃ij gkl (x, t),
∂xp
¸
·
p q
kl
k pr qs ∂gpq
k pr qs ∂gpq
= − 2g gpq Γik Γjl + gik Γrs g g
+ gjk Γrs g g
∂xj
∂xi
³
´
³
´
1
= −2 δ kl + H ij (δpq + hpq ) (δ pa + H pa ) δ qb + H qb
4µ
µ
¶
¶
∂hbj
∂hai ∂hak
∂hik
∂hbl ∂hjl
·
+
−
+
−
∂xi
∂xa
∂xj
∂xk
∂xl
∂xb
− (δik + Hik ) (δ pr + H pr ) (δ qs + H qs )
¶
´ µ ∂h
1 ³ ka
∂has ∂hrs ∂hpq
ar
ka
· δ +H
+
−
2
∂xs
∂xr
∂xa
∂xj
pr
pr
qs
qs
− (δjk + Hjk ) (δ + H ) (δ + H )
¶
´ µ ∂h
1 ³ ka
∂has ∂hrs ∂hpq
ar
ka
· δ +H
+
−
2
∂xs
∂xr
∂xa
∂xi
µ
¶µ
¶
∂hbj
∂hai ∂hak
∂hik
∂hbl ∂hjl
1
+
−
+
−
= −2 · · δ kl δpq δ pa δ qb
4
∂xi
∂xa
∂xj
∂xk
∂xl
∂xb
µ
¶
1
∂har
∂has ∂hrs ∂hpq
− δik δ pr δ qs δ ka
+
−
s
2
∂x
∂xr
∂xa
∂xj
¯
¯¶
µ
¶
µ
¯ ∂hkl ¯ 3
1
∂has ∂hrs ∂hpq
pr qs ka ∂har
− δjk δ δ δ
+
−
+ O |hkl | + ¯¯ p ¯¯
2
∂xs
∂xr
∂xa
∂xi
∂x
¶µ
¶
µ
∂hbj
∂hbl ∂hjl
∂hik
1
∂hai ∂hak
= − δ kl δ ab
+
−
+
−
i
a
k
l
2
∂x
∂x
∂xj
∂x
∂x
∂xb
µ
¶
∂hir
∂his ∂hrs ∂hpq
1
+
−
− δ pr δ qs
s
2
∂x
∂xr
∂xi
∂xj
¯
¯¶
¶
µ
µ
¯ ∂hkl ¯ 3
∂hjs ∂hrs ∂hpq
1 qs pr ∂hjr
¯
− δ δ
+
−
+ O |hkl | + ¯ p ¯¯ .
2
∂xs
∂xr
∂xj
∂xi
∂x
∂hkl (x, t)
δkl + hkl (x, t),
∂xp
¶
µ
(4.2)
348
Wen-Rong Dai, De-Xing Kong, Kefeng Liu
Furthermore, (4.2) can be written as

µ
¶
n
X
∂ 2 hij

∂2
∂hkl ∂ 2 hkl


,
,
+ H̄ij hkl ,
 ∂t2 hij (x, t) =
∂xp ∂xp ∂xq
∂xk ∂xk
k=1



hij (x, 0) = εg 0 (x), ∂ hij (x, 0) = εg 1 (x),
ij
ij
∂t
where
µ
µ
¶
¶
∂ 2 hij
∂hkl
∂hkl ∂ 2 hkl
kl
e
+ Hij δkl + hkl ,
H̄ij hkl ,
,
=H
∂xp ∂xp ∂xq
∂xp
∂xk xl
µ
¶
¶µ
∂hbj
∂hbl ∂hjl
1 kl ab ∂hai ∂hak
∂hik
+
+
=− δ δ
−
−
2
∂xi
∂xa
∂xj
∂xk
∂xl
∂xb
µ
¶
1
∂hir
∂his ∂hrs ∂hpq
− δ pr δ qs
+
−
s
2
∂x
∂xr
∂xi
∂xj
µ
¶
∂hjr
∂hjs ∂hrs ∂hpq
1
− δ pr δ qs
+
−
s
2
∂x
∂xr
∂xj
∂xi
¯
¯ ¯
¯¶
µ
¯ ∂hkl ¯ ¯ ∂ 2 hkl ¯ 3
∂ 2 hij
kl
¯
¯
¯
−h
+ O |hkl | + ¯ p ¯ + ¯ p q ¯¯ .
∂x
∂x ∂x
∂xk ∂xl
Let
µ
λ̂ =
∂hkl ∂ 2 hkl
hkl ,
,
∂xp ∂xp ∂xq
The nonlinear term
µ
H̄ij (λ̂) = H̄ij
(4.3)
¶
(p, q, k, l = 1, 2 · · · , n).
∂hkl ∂ 2 hkl
hkl ,
,
∂xp ∂xp ∂xq
¶
in (4.3) is smooth in a neighborhood about λ̂ = 0 and satisfies
´
³
(i, j = 1, 2, · · · , n).
H̄ij (λ̂) = O |λ̂|2
By the well-known global existence results for the nonlinear wave equation (e.g.,
see [2], [7], [8], [10]), there exists a unique global smooth solution (hij (x, t)) for
the Cauchy problem (4.3) or (4.2). Thus, the proof of Theorem 4.1 is complete.
¶
Remark 4.2. For general complete noncompact Riemannian manifold M , under
suitable bounded assumptions we can prove that any Ricci flat Riemannian metric
g ij (x) on M possesses the locally nonlinear stability. For the globally nonlinear
stability, when the space dimension n is large that 4, i.e., n ≥ 5, under suitable assumptions, we can show that the Ricci flat Riemannian metric g ij (x) on
Hyperbolic Geometric Flow (I)...
349
M possesses the globally nonlinear stability, however when n = 3, 4, this result
in general is not true, since in this situation the solution to the corresponding
nonlinear wave equation in general blows up in finite time.
5. Wave character of the curvatures
The hyperbolic geometric flow is a system of hyperbolic evolution equations
on the metrics. The evolution of the metrics implies a system of nonlinear wave
equations for the Riemannian curvature tensor Rijkl , the Ricci curvature tensor
Rij and the scalar curvature R which we will derive.
Let M be an n-dimensional complete manifold. We consider the hyperbolic
geometric flow on M , that is,
∂2
gij (x, t) = −2Rij (x, t).
(5.1)
∂t2
We now want to find the evolution equations for the Riemannian curvature tensor
Rijkl , the Ricci curvature tensor Rij and the scalar curvature R.
Direct computations yield
Γhjl
1
= g hm
2
µ
µ
∂gmj
∂gml
∂gjl
+
− m
l
j
∂x
∂x
∂x
¶
,
¶
µ
¶
∂ 2 gmj
∂ 2 gml
∂ 2 gjl
∂gml
∂gjl
1 ∂g hm ∂gmj
+
−
+
−
+
,
∂xl ∂t
∂xj ∂t ∂xm ∂t
2 ∂t
∂xl
∂xj
∂xm
µ
¶
µ
¶
1 ∂ 2 g hm ∂gmj
∂gml
∂gjl
1 ∂g hm ∂ 2 gmj
∂ 2 gml
∂ 2 gjl
∂2 h
Γ
=
+
−
+
2
·
+
−
∂t2 jl 2 ∂t2
∂xl
∂xj
∂xm
2 ∂t
∂xl ∂t
∂xj ∂t ∂xm ∂t
µ
µ 2
¶
µ 2
¶
µ 2 ¶¶
1
∂
∂ gmj
∂
∂ gml
∂
∂ gjl
+ g hm
+ j
− m
,
2
∂xl
∂t2
∂x
∂t2
∂x
∂t2
∂ h
1
Γjl = g hm
∂t
2
∂Γhjl
∂Γhil
−
+ Γhip Γpjl − Γhjp Γpil ,
i
∂x
∂xj
µ 2
¶
µ 2
¶
´
∂
∂
∂2 ³ h p
∂2 h
∂ h
∂ h
h p
R
=
Γ
−
Γ
+
Γ
Γ
−
Γ
Γ
ip jl
jp il ,
∂t2 ijl ∂xi ∂t2 jl
∂xj ∂t2 il
∂t2
2
¢
∂2
∂2 ¡
∂2 h
∂ghk ∂ h
h
h ∂ ghk
Rijkl = 2 ghk Rijl
= ghk 2 Rijl
+ Rijl
+2
R
2
2
∂t
∂t
∂t
∂t
∂t ∂t ijl
·
µ 2
¶
µ 2
¶
´¸
∂
∂ h
∂ h
∂
∂2 ³ h p
h p
= ghk
Γ
Γ
− j
+ 2 Γip Γjl − Γjp Γil
∂xi ∂t2 jl
∂x
∂t2 il
∂t
µ
µ
·
¶
¶
´¸
∂ghk ∂
∂ h
∂ h
∂
∂ ³ h p
h p
+2
Γip Γjl − Γjp Γil
Γ
− j
Γ
+
∂t ∂xi ∂t jl
∂x
∂t il
∂t
h
Rijl
=
h
+Rijl
∂ 2 ghk
.
∂t2
350
Wen-Rong Dai, De-Xing Kong, Kefeng Liu
We choose the normal coordinates around a fixed point p on M such that
Γkij (p) = 0,
or, equivalently,
∂gij
(p) = 0.
∂xk
Then we have
∂2
Rijkl =
∂t2
µ
¶¸
·
µ
¶
∂ 1 ∂ 2 g hm ∂gmj
∂gml
∂gjl
1 ∂g hm ∂ 2 gmj
∂ 2 gml
∂ 2 gjl
ghk i
+
−
+
2
·
+
−
∂x 2 ∂t2
∂xl
∂xj
∂xm
2 ∂t
∂xl ∂t
∂xj ∂t ∂xm ∂t
·
µ
µ 2
¶
µ 2
¶
µ 2 ¶¶¸
∂
∂ gmj
∂
∂ gml
∂
∂ gjl
∂ 1 hm
g
+ j
− m
+ghk i
∂x 2
∂xl
∂t2
∂x
∂t2
∂x
∂t2
·
µ
¶
µ
¶¸
∂ 1 ∂g hm ∂gmi
∂ 2 gml
∂ 2 gil
∂gml
∂gil
1 ∂g hm ∂ 2 gmi
−ghk j
+
−
+
−
+
2
·
∂x 2 ∂t2
∂xl
∂xi
∂xm
2 ∂t
∂xl ∂t
∂xi ∂t
∂xm ∂t
µ
·
µ 2
¶
µ 2
¶
µ 2 ¶¶¸
∂
∂ 1 hm
∂ gmi
∂
∂ gml
∂
∂ gil
−ghk j
g
+ i
− m
∂x 2
∂xl
∂t2
∂x
∂t2
∂x
∂t2
¶
µ
∂
∂
∂ghk
∂ h ∂ p
Γip Γjl − Γhjp Γpil + 2
+2ghk
∂t
∂t
∂t
∂t
∂t
·
µ hm µ
¶¶
µ hm µ
¶¶¸
1 ∂
∂gml
∂gjl
∂gml
∂gil
∂g
∂gmj
1 ∂
∂g
∂gmi
·
+
− m
+
− m
−
2 ∂xi
∂t
∂xl
∂xj
∂x
2 ∂xj
∂t
∂xl
∂xi
∂x
µ
µ
¶
µ
¶
µ
¶¶
∂
∂gmj
∂
∂gml
∂
∂gjl
∂ghk 1 hm ∂
g
+ j
− m
+2
i
l
∂t 2
∂x ∂x
∂t
∂x
∂t
∂x
∂t
µ
µ
¶
µ
¶
µ
¶¶
2
∂ghk 1 hm ∂
∂
∂gmi
∂
∂gml
∂
∂gil
h ∂ ghk
−2
g
+
−
+ Rijl
j
l
i
m
∂t 2 · ∂x µ ∂x
∂t
∂x ¶ ∂t
∂t2 ¶¶¸
µ ∂xhm µ ∂t
2 hm
1
∂ g
∂
∂gmj
∂gml
∂gjl
∂g
∂gmi
∂gml
∂gil
∂
+
− m − j
+
− m
= ghk
2
∂t2
∂xi
∂xl
∂xj
∂x
∂x
∂t
∂xl
∂xi
∂x
µ
¶
µ
¶
∂ 2 g hm ∂gmj
∂gml
∂gjl
∂ 2 g hm ∂gmi
∂gml
∂gil
+ghk
+
−
− ghk j
+
−
∂xi ∂t ∂xl ∂t ∂xj ∂t ∂xm ∂t
∂x ∂t ∂xl ∂t ∂xi ∂t ∂xm ∂t
·
µ
¶
µ 2
¶¸
∂gmj
∂gml
∂gjl
∂
∂ gmi
∂ 2 gml
∂ 2 gil
∂g hm ∂
+ghk
+
−
− j
+
− m
∂xi ∂x¶l ∂t ∂xj ∂tµ ∂xm¶∂t
∂x µ ∂xl ∂t¶¸ ∂xi ∂t
∂x ∂t
· ∂t2 µ
2
2
2
2
2
∂ gkj
∂
∂ gkl
∂
∂ gjl
1
∂
+ i j
− i k
+
2 ∂xi ∂xl
∂t2
∂x ∂x
∂t2
∂x ∂x
∂t2
µ
¶
µ
¶
µ
¶¸
·
1
∂ 2 gki
∂2
∂ 2 gkl
∂2
∂ 2 gil
∂2
−
+
−
j
l
2 ∂x
∂t2
∂xj ∂xi ¶∂t2
∂xj ∂xk
∂t2
µ ∂x
∂ h ∂ p
∂
∂ p
+2ghk
Γ
Γ − Γh
Γ
∂t ip ∂t jl ∂t jp ∂t il
Hyperbolic Geometric Flow (I)...
351
·
µ
¶
µ
¶¸
∂gjl
∂gmj
∂ghk ∂g hm ∂
∂gml
∂gil
∂
∂gmi ∂gml
+
+
+
− m − j
− m
∂t ∂t
∂xi
∂xj
∂x
∂x
∂xi
∂x
∂xl
∂xl
·
µ
¶
µ
¶¸
∂gjl
∂gmj
∂gmi
∂gil
∂ghk hm ∂
∂gml
∂
∂gml
g
−
−
+
+
− j
+
∂t
∂xi ∂xl ∂t ∂xj ∂t ∂xm ∂t
∂x
∂xl ∂t ∂xi ∂t ∂xm ∂t
h
+Rijl
∂ 2 ghk
.
∂t2
(5.2)
Noting g hm gml = δlh , we get
∂gpq
∂g hm
= −g hp g mq
,
∂t
∂t
∂gpq
∂ 2 g hm
= −g hp g mq k ,
∂xk ∂t
∂x ∂t
2
∂ 2 g hm
hp g mq ∂ gpq + 2g hp g rq g sm ∂gpq ∂grs .
=
−g
∂t2
∂t2
∂t ∂t
Thus, it follows from (5.2) that
µ
¶
∂gkq ∂grp
∂2
1 pm ∂ 2 gkp
rq
pm
Rijkl = − g
+g g
∂t2
2
∂t2
∂t ∂t
·
µ
¶
µ
¶¸
2
∂gjl
∂gmj
∂
∂gml
∂
∂gmi ∂gml
∂gil
pm ∂ gkp
×
+
−
−
+
−
−
g
∂xi
∂xj
∂xm
∂xj
∂xi
∂xm
∂xi ∂t
∂xl
∂xl
µ 2
¶
µ
¶
∂ 2 gjl
∂ 2 gkp ∂ 2 gmi ∂ 2 gml
∂ gmj
∂ 2 gml
∂ 2 gil
pm
·
+ j − m
+
+g
−
∂x ∂t ∂x ∂t
∂xj ∂t ∂xl ∂t
∂xi ∂t ∂xm ∂t
∂xl ∂t
·
µ 2 ¶
µ 2 ¶
µ 2 ¶¸
∂ gkj
∂ gjl
1
∂2
∂2
∂ gkl
∂2
+
+ i j
− i k
2
2
i
l
2 ∂x ∂x
∂t
∂x ∂x
∂t
∂t2
∂x ∂x
·
µ
¶
µ
¶
µ
¶¸
1
∂2
∂ 2 gki
∂2
∂ 2 gkl
∂2
∂ 2 gil
−
+ j i
− j k
2 ∂xj ∂xl
∂t2
∂x ∂x
∂t2
∂t2
∂x ∂x
¶
µ
2
∂
∂
∂ h ∂ p
h ∂ ghk − g hp g mq ∂gnk ∂gpq
+ 2ghk
Γip · Γjl − Γhjp · Γpil + Rijl
∂t
∂t
∂t
∂t
∂t2
∂t ∂t
·
µ
¶
µ
¶¸
∂gjl
∂gmj
∂
∂gml
∂
∂gmi ∂gml
∂gil
·
+
− m − j
+
− m
∂xi µ∂xl
∂xj
∂x
∂x
∂x
∂xl µ ∂xi
¶
¶
∂ 2 gkp
∂g
∂g
∂
∂ p
∂ p
∂ p
rp
kq
p
rq
=−
Γ −
Γ
+ 2g
Γ −
Γ
∂t2
∂xi jl ∂xj il
∂t ∂t
∂xi jl ∂xj il
¶
µ
∂ 2 gjl
∂ 2 gkp pm ∂ 2 gmj
∂ 2 gml
+ j − m
− i ·g
∂x ∂t
∂x ∂t ∂x ∂t
∂xl ∂t
¶
µ 2
2
∂ gkp pm ∂ gmi ∂ 2 gml
∂ 2 gil
+ j ·g
+
−
∂x ∂t
∂xi ∂t ∂xm ∂t
∂xl ∂t
Wen-Rong Dai, De-Xing Kong, Kefeng Liu
352
·
¸
1
∂2
∂2
∂2
+
(−2Rkj ) + i j (−2Rkl ) − i k (−2Rjl )
2 ∂xi ∂xl
∂x ∂x
∂x ∂x
¸
·
1
∂2
∂2
∂2
−
(−2Rik ) + i j (−2Rkl ) − j k (−2Ril )
2 ∂xj ∂xl
∂x ∂x
∂x ∂x
·
¸
∂ h ∂ p
∂
∂
∂ghk ∂gpq hp
+2ghk
Γ · Γ − Γh · Γp −
g
∂t ip ∂t jl ∂t jp ∂t il
∂t ∂t
µ
¶
2
∂ q
∂ q
h ∂ ghk
·2
Γjl −
Γil + Rijl
∂xi
∂xj
∂t2
¸
·
∂2
1
∂2
∂2
(−2Rkj ) + i j (−2Rkl ) − i k (−2Rjl )
=
2 ∂xi ∂xl
∂x ∂x
∂x ∂x
·
¸
2
2
2
1
∂
∂
∂2
pm ∂ gkp
−
(−2R
)
+
(−2R
)
−
(−2R
)
−
g
ki
il
kl
2 ∂xj ∂xl
∂xi ∂xj
∂xi ∂t
∂xj ∂xk
¶
µ
¶
µ 2
∂ 2 gjl
∂ 2 gkp ∂ 2 gmi ∂ 2 gml
∂ gmj
∂ 2 gil
∂ 2 gml
pm
+g
−
·
+ j − m
+
∂x ∂t ∂x ∂t
∂xj ∂t ∂xl ∂t
∂xi ∂t ∂xm ∂t
∂xl ∂t
µ
¶
∂ h ∂ p
∂ h ∂ p
+2ghk
Γ · Γ − Γ · Γ .
∂t ip ∂t jl ∂t jp ∂t il
(5.3)
On the one hand, we have
∂2
Rjk = ∇i ∇l Rjk + ∇i Γplk · Rjp + ∇i Γplj · Rkp .
∂xi ∂xl
Then
·
¸
1
∂2
∂2
∂2
(−2Rkj ) + i j (−2Rkl ) − i k (−2Rjl )
2 ∂xi ∂xl
∂x ∂x
∂x ∂x
·
¸
2
2
1
∂
∂2
∂
−
(−2Rki ) + i j (−2Rkl ) − j k (−2Ril )
2 ∂xj ∂xl
∂x ∂x
∂x ∂x
= −∇i ∇l Rkj − ∇i Γplk Rjp − ∇i Γplj Rkp − ∇i ∇j Rkl − ∇i Γpjk Rlp − ∇i Γpjl Rkp
+∇i ∇k Rjl + ∇i Γpkj Rlp + ∇i Γpkl Rjp + ∇i ∇l Rki + ∇j Γpkl Rip + ∇j Γpli Rkp
+∇j ∇i Rkl + ∇j Γpik Rlp + ∇j Γpil Rkp − ∇j ∇k Ril − ∇j Γpki Rpl − ∇j Γpkl Rpi
= −∇i ∇l Rkj − ∇i ∇j Rkl + ∇i ∇k Rjl + ∇j ∇l Rki + ∇j ∇i Rki − ∇j ∇k Ril
¡
¢
¡
¢
+Rip ∇j Γplk − ∇j Γpkl + Rjp −∇i Γplk + ∇i Γpkl
³
´
+Rkp −∇i Γplj − ∇i Γpjl + ∇j Γpli + ∇j Γpil
Hyperbolic Geometric Flow (I)...
353
´
³
+Rlp −∇i Γpjk + ∇i Γpkj + ∇j Γpik − ∇j Γpki
= −∇i ∇l Rkj + ∇i ∇k Rjl + ∇j ∇l Rki − ∇j ∇k Ril
´
³
p
−Rijlp g pq Rqk − Rijkp g pq Rql + Rkp −2Rijl
(5.4)
= −∇i ∇l Rkj + ∇i ∇k Rjl + ∇j ∇l Rki − ∇j ∇k Ril − g pq (Rijlp Rkq + Rijkp Rlq ) .
On the other hand, we have
µ
¶
2
∂ 2 gjl
∂ 2 gkp ∂ 2 gmj
∂ 2 gml
pm ∂ gkp
pm
−
+
g
+
−g
∂xi ∂t ∂xl ∂t
∂xj ∂t ∂xm ∂t
∂xj ∂t
µ 2
¶
µ
¶
∂ gmi ∂ 2 gml
∂ 2 gil
∂ h ∂ p
∂ h ∂ p
·
+
−
+ 2ghk
Γ · Γ − Γ · Γ
∂xi ∂t ∂xm ∂t
∂t ip ∂t jl ∂t jp ∂t il
∂xl ∂t
µ
¶
∂ 2 gkp ∂ 2 gmj
∂ 2 gjl
∂ 2 gml
pm
= −g
+ j − m
∂xi ∂t ∂xl ∂t
∂x ∂t ∂x ∂t
µ 2
¶
2
∂ gkp ∂ gmi ∂ 2 gml
∂ 2 gil
pm
+
−
+g
∂xj ∂t ∂xl ∂t
∂xi ∂t ∂xm ∂t
µ 2
¶
¶µ 2
∂ 2 gkp
∂ 2 gjl
∂ 2 gip
∂ gmj
1
∂ gki
∂ 2 gml
+ g pm
+
−
−
+
2
∂xp ∂t ∂xi ∂t ∂xk ∂t
∂xj ∂t ∂xm ∂t
∂xl ∂t
µ 2
¶
¶
µ
∂ 2 gkp
∂ 2 gjp
1 pm ∂ gkj
∂ 2 gil
∂ 2 gmi ∂ 2 gml
(5.5)
− g
+
−
−
+
2
∂xp ∂t ∂xj ∂t ∂xk ∂t
∂xi ∂t ∂xm ∂t
∂xl ∂t
½ µ 2
¶µ 2
¶
∂ 2 gjl
∂ 2 gkp
∂ 2 gip
1 ∂ gmj
∂ 2 gml
∂ gki
pm
=g
+ j − m
−
−
2 ∂xl ∂t
∂x ∂t ∂x ∂t
∂xp ∂t ∂xk ∂t ∂xi ∂t
µ
¶µ 2
¶¾
∂ gkj
∂ 2 gkp
∂ 2 gjp
1 ∂ 2 gmi ∂ 2 gml
∂ 2 gil
−
+
−
−
−
2 ∂xl ∂t
∂xi ∂t ∂xm ∂t
∂xp ∂t ∂xk ∂t ∂xj ∂t
½
µ
¶
µ
¶
1 pr ∂ ∂grj
1 qs ∂ ∂gki ∂gis ∂gks
∂grl ∂gjl
= 2gpq
g
−
· g
−
+
−
2
∂t ∂xl
∂xj
∂xr
2
∂t ∂xs
∂xi
∂xk
µ
¶
µ
¶¾
∂gjs ∂gks
1
∂ ∂gri ∂grl ∂gil
1
∂ ∂gkj
− g pr
−
· g qs
−
+
−
i
r
s
l
2
∂t ∂x
∂x
∂x
2
∂t ∂x
∂xj
∂xk
µ
¶
∂ p ∂ q
∂
∂
= 2gpq
Γil · Γjk − Γpjl · Γqik .
∂t
∂t
∂t
∂t
Therefore, it follows from (5.3), (5.4) and (5.5) that
∂2
Rijkl = − ∇i ∇l Rkj + ∇i ∇k Rjl + ∇j ∇l Rki − ∇j ∇k Ril
∂t2
µ
¶
∂ p ∂ q
∂ p ∂ q
pq
− g (Rijql Rkp + Rijkq Rkp ) + 2gpq
Γ · Γ − Γ · Γ
.
∂t il ∂t jk ∂t jl ∂t ik
Wen-Rong Dai, De-Xing Kong, Kefeng Liu
354
Similar to Hamilton [6], we have
Theorem 5.1. Under the hyperbolic geometric flow (5.1), the Riemannian curvature tensor Rijkl satisfies the evolution equation
∂2
Rijkl = 4Rijkl + 2 (Bijkl − Bijlk − Biljk + Bikjl )
∂t2
−g pq (R
µpjkl Rqi + Ripkl Rqj + Rijpl Rqk ¶+ Rijkp Rql )
∂ p ∂ q
∂
∂
+2gpq
Γ · Γ − Γp · Γq ,
∂t il ∂t jk ∂t jl ∂t ik
(5.6)
where Bijkl = g pr g qs Rpiqj Rrksl and 4 is the Laplacian with respect to the evolving
metric.
Remark 5.1. In Theorem 5.1 and Theorem 5.2 below, the term
µ
¶
∂ p ∂ q
∂ p ∂ q
Γ · Γ − Γ · Γ
2gpq
∂t il ∂t jk ∂t jl ∂t ik
can be written in the covariant form. For the sake of simplicity, we omit it.
For the Ricci curvature tensor, we have
´
∂2
∂2 ³
jl
R
=
R
g
ik
ijkl
∂t2
∂t2
∂2
∂
∂
∂ 2 g jl
= g jl 2 Rijkl + 2 g jl · Rijkl + Rijkl
∂t
∂t
∂t
∂t2
2
∂gpq ∂
∂ 2 gpq
∂
Rijkl − g jp g lq
Rijkl
= g jl 2 Rijkl − 2g jp g lq
∂t
∂t ∂t
∂t2
∂gpq ∂grs
+ 2g jp g rq g sl
Rijkl .
∂t ∂t
Thus, we obtain
Theorem 5.2. Under the hyperbolic geometric flow (5.1), the Ricci curvature
tensor satisfies
∂2
Rik = 4Rik + 2g pr g qs Rpiqk Rrs − 2g pq Rpi Rqk
∂t2
µ
¶
∂ p ∂ q
∂ p ∂ q
jl
+2g gpq
Γ
Γ − Γ
Γ
∂t il ∂t jk ∂t jl ∂t ik
∂gpq ∂grs
∂gpq ∂
Rijkl + 2g jp g rq g sl
Rijkl .
−2g jp g lq
∂t ∂t
∂t ∂t
(5.7)
Hyperbolic Geometric Flow (I)...
355
For the scalar curvature, we have
´
∂2
∂ 2 ³ ik
R
=
g
R
ik
∂t2
∂t2
∂2
∂
∂
∂ 2 g ik
= g ik 2 Rik + 2 Rik · g ik + Rik
∂t
∂t
∂t
∂t2
µ
¶
2
2
ip kq ∂ gpq
ik ∂
ip kq ∂gpq ∂Rik
ip rq sk ∂gpq ∂grs
=g
+ Rik −g g
.
Rik − 2g g
+ 2g g g
∂t2
∂t ∂t
∂t2
∂t ∂t
On the other hand,
µ
¶
∂ p ∂ q
∂ p ∂ q
2g g gpq
Γ
Γ − Γ
Γ
∂t il ∂t jk ∂t jl ∂t ik
∂gjl
∂gij
3
∂gkl
1
∂gik
= g ik g jl g rs ∇r (
)∇s (
) − g rs ∇r (g ik
)∇s (g jl
)
2
∂t
∂t
2
∂t
∂t
∂gjs
∂gij
∂gik
∂gks
+2g rs g jl ∇r (g ik
)∇l (
) − g ik g jl g rs ∇r (
)∇l (
)
∂t
∂t
∂t
∂t
∂gkr
∂gls
− 2g ik g jl g rs ∇i (
)∇j (
).
∂t
∂t
ik jl
Then, we get
Theorem 5.3. Under the hyperbolic geometric flow (5.1), the scalar curvature
satisfies
∂2
R = 4R + 2|Ric|2
∂t2
∂gjl
∂gij
∂gkl
∂gik
)∇s (
) − 12 g rs ∇r (g ik
)∇s (g jl
)
∂t
∂t
∂t
∂t
∂gjs
∂gij
∂gks
∂gik
+2g rs g jl ∇r (g ik
)∇l (
) − g ik g jl g rs ∇r (
)∇l (
) (5.8)
∂t
∂t
∂t
∂t
∂gpq ∂
∂gkr
∂gls
−2g ik g jl g rs ∇i (
)∇j (
) − 2g ik g jp g lq
Rijkl
∂t
∂t
∂t ∂t
∂gpq ∂grs
∂gpq ∂Rik
−2g ip g kq
+ 4Rik g ip g rq g sk
.
∂t ∂t
∂t ∂t
+ 32 g ik g jl g rs ∇r (
Theorems 5.1-5.3 show that the curvatures of the hyperbolic geometric flow
possess the wave character. We will apply techniques from hyperbolic equations
to the above wave equations of curvatures to derive various geometric results.
356
Wen-Rong Dai, De-Xing Kong, Kefeng Liu
6. Discussions
The hyperbolic geometric flow describes the wave character of the metrics and
curvatures of manifolds. Many hyperbolic systems in nature provide natural
singular sets, the typical example is the Einstein equations in general relativity
which form a hyperbolic system with a well-posed Cauchy problem. If one starts
with smooth initial data, one may end up with a singular space-time. One of the
most challenging problems is to describe the kind of natural singularity. The famous cosmic censorship conjecture due to Penrose is an attempt to describe such
singularities (see [12]). In Kong and Liu [11], we construct some exact solutions
of the hyperbolic geometric flow, these solutions possess the singularities which
are nothing but those described by Penrose’s conjecture. By these examples, we
believe that the hyperbolic geometric flow is a very natural and powerful tool to
understand the singularities in the nature, in particular, the singularity described
by Penrose cosmic censorship conjecture.
The Einstein equations play an essential role in general relativity. Consider a
space-time with Lorentzian metric
ds2 = gµν dxµ dxν
(µ, ν = 0, 1, 2 · · · , n).
(6.1)
The vacuum Einstein equations read
Gµν = 0,
(6.2)
where Gµν is the Einstein tensor. We now consider the following metric with
orthogonal time-axis
ds2 = −dt2 + gij (x, t)dxi dxj .
(6.3)
Substituting (6.3) into (6.2), we can obtain the equations satisfied by the metric
gij
∂ 2 gij
∂gij ∂gpq
∂gip ∂gjq
1
= −2Rij − g pq
+ g pq
.
(6.4)
2
∂t
2
∂t ∂t
∂t ∂t
Neglecting the lower order terms gives the hyperbolic geometric flow (1.1). Therefore, in this sense, the hyperbolic geometric flow can be viewed as the leading
terms in the vacuum Einstein equations with respect to the metric (6.3). Since
the hyperbolic geometric flow only contains the main terms in the Einstein equations, it not only becomes simpler and more symmetric, but also possesses rich
and beautiful geometric properties. In particular, in mathematics, its Cauchy
problem is well-posed and easier to handle some fundamental problems such as
Hyperbolic Geometric Flow (I)...
357
the global existence and formation of singularities; on the other hand, it can
be applied to re-understand the singularity of the universe and other important problems in physics and cosmology (see [16]). We also believe that there
should be some relations between the solutions of the Einstein equations and the
corresponding hyperbolic geometric flows. On the other hand, from the above
discussions we have seen that the hyperbolic geometric flow also possesses many
beautiful features similar to those of the Ricci flow, and some of the techniques
in the study of the Ricci flow can be directly used to understand the hyperbolic
geometric flow. The deep study on the hyperbolic geometric flow may open a
new way to understand the complicated Einstein equations.
It is well known, in general relativity there is a constraint system of equations
involving an asymptotically flat metric tensor and another symmetric tensor.
There are four constraint equations and it is therefore over-determined. Unlike
this, since the time axis is orthogonal to other space axes, the hyperbolic geometric flow does not need to satisfy any additional constraint. More precisely,
for the Cauchy problem of the hyperbolic geometric flow, in order to determine
the solution we need two initial conditions: one is the metric flow itself gij (x, 0),
∂g
another is its derivative ∂tij (x, 0), since the time axis is orthogonal to other axes,
these initial data do not need to satisfy any additional constraint, and therefore
it is a determined system. This is another main new feature of the hyperbolic
geometric flow.
Many mathematicians, for example Shatah et al [13]-[15], have investigated
the Cauchy problem for some geometric wave equations. The model at hand
is the harmonic map problem, which is the study of maps from the Minkowski
space-time into complete Riemannian manifolds. This kind of geometric wave
equations is a system of partial differential equations of second order, which is the
Euler-Lagrange equations of the action integral of the harmonic map. It satisfies
certain linear matching condition, and then under suitable assumptions, has a
unique small smooth solution for all time, and possesses some interesting (decay,
energy and regularity) estimates. On the other hand, the hyperbolic geometric
flow is determined by the Ricci curvatures of a family of Riemannian metrics on
the manifold under consideration. That is to say, the hyperbolic geometric flow
possesses itself intrinsic geometric structure and can be used to describe the wave
character of metrics and curvatures. This is essentially different from the above
harmonic map problem.
358
Wen-Rong Dai, De-Xing Kong, Kefeng Liu
As well-known, one can understand the heat kernel from the kernel of wave
equation. This indicates that we should be able to derive various information of
the Ricci flow from that of the hyperbolic geometric flow. Therefore it is also
interesting to understand the relations between the hyperbolic geometric flow
and the Ricci flow, the singularities of its solutions and its relation with the
geometrization theorem. This will be another interesting topic in the sequel.
Acknowledgements.
The authors thank the referee for valuable suggestions. The work of Dai was supported by the fund of post-doctor of China
(Grant No. 20060401063); the work of Kong was supported in part by the NNSF
of China (Grant No. 10671124) and the Qiu-Shi Professor Fellowship from Zhejiang University; the work of Liu was supported in part by the NSF and NSF of
China.
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Wen-Rong Dai, De-Xing Kong
Center of Mathematical Sciences, Zhejiang University
Hangzhou 310027, China
E-mail: [email protected].
Kefeng Liu
Center of Mathematical Sciences, Zhejiang University
Hangzhou 310027, China
or
Department of Mathematics, UCLA, CA 90095, USA
E-mail: [email protected]