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ANALYSIS ASPECTS OF RICCI FLOW ON CONICAL
SURFACES
HAO YIN

Abstract
...
In particular, we prove the long time existence of the conical Ricci flow
for general cone angle and show that this solution has the optimal regularity,
namely, the time derivatives of the conformal factor are bounded and for each
fixed time, the conformal factor has an explicit asymptotic expansion near the
cone points
...
Introduction
2
...
1
...
2
...
3
...
4
...
5
...
Smoothing effect of linear equation
3
...
H¨lder regularity for bounded solution
o
3
...
Smoothing estimate for rough initial data
4
...
1
...
2
...
3
...
Higher regularity of conical Ricci flow
5
...
Regularity of ∂t K
5
...
Higher order regularity
6
...
1
...
2
...
3
...
Proof of Theorem 3
...
1
...
2
...
1
Appendix B
...
10
Appendix C
...
11
Appendix D
...

1

2
5
5
8
10
12
15
16
16
19
25
25
30
33
34
35
37
37
38
39
41
44
45
46
47
50
50
52

2

HAO YIN

1
...
Let S be a smooth Riemann surface and {pi } be finitely many prescribed
points on S
...
We are interested in the class
of metrics g which are smooth and compatible with the conformal structure of S
away from pi while having a conical singularity of order βi at pi , i
...
in a conformal
coordinate chart U around pi , g is given by
e2u r2β (dx2 + dy 2 ),
where r = x2 + y 2 and u is at least continuous around pi
...
For example, the conical singularities can be smoothed
out immediately to become the ordinary Ricci flow on closed manifolds as in [19,
18, 20], or the conical singularities can be pushed to infinity immediately to become
some Ricci flow on complete noncompact surfaces as in [21]
...

To describe the flow, it is convenient to have a background metric g which is
˜
exactly the cone metric of order βi near small neighborhood of each pi
...
Consider the family of metrics g(t) given by e2u(t) g with u(t) being
˜
‘good’ near pi so that g(t) is still conical at pi and u(t) satisfies the equation
(1
...
The exact meaning of ‘being good’ depends on the
approach we take to study the problem and is also a central theme of the present
paper
...
The study was continued in [25] which proves
the long time existence of the flow if βi ∈ (−1, 0) and gives some convergence results
in certain cases
...
We
refer the interested readers to the historical remark near the end of this section
...
Mazzeo, Rubinstein and Sesum [14], backed up by the
mircolocal analysis method developed by Mazzeo [13], Bahuaud and Vertman [1],
Jeffres and Loya [8], Mooers [15] and others, proved the long time existence of (1
...
This last phenomenon, which
is called the one dimensional Hamilton-Tian conjecture by the authors in [14], was
proved by Phong, Song, Strum and Wang [16, 17]
...
e
...
They developed in [3] a parabolic version of Donaldson’s
C 2,α estimate [4]
...


ANALYSIS ASPECTS OF RICCI FLOW ON CONICAL SURFACES

3

In this paper, we discuss the PDE aspect of the conical Ricci flow by developing
a weak solution theory to the linear parabolic equation on surfaces with conical
singularities
...
We refer to Section 2 and Section 3 for the
exact statement of these results
...

The first is an existence result
...
1
...

˜
For any function u0 : S → R satisfying that u0 and ˜ u0 lie in W 2,α , there exist
T > 0 depending only on the W 2,α norm of u0 and ˜ u0 , and u(t) ∈ V 2,α,[0,T ]
solving (1
...
If r is chosen so that r/2 is the average of Gauss curvature of the
metric e2u0 g , then u(t) is defined on [0, ∞) and for any T ∈ [0, ∞), u ∈ V 2,α,[0,T ]
...
It suffices for
now to remark that functions in these spaces are C 2,α away from the singularities
and have some bounded integral norms, putting restrictions near the singularities
so that they behave like the weak solutions in the standard theory of parabolic
equation on smooth domains (see Chapter III of [10] for example)
...
1 holds for any βi > −1, while the long time existence results in [25]
and [14] are both restricted to the case of sharp cone angles, i
...
βi ∈ (−1, 0) (note
that this is equivalent to βi ∈ (0, 1) in [14])
...
The proof here focuses on the
conformal factor u instead of the curvature K
...

Given the existence of the solution in Theorem 1
...
Usually, the higher order regularity
of a nonlinear PDE is proved by successively taking derivatives
...

Theorem 1
...
Let u(t) be the solution given in Theorem 1
...
For all k ∈ N and
k
k
0 < δ < T < ∞, ∂t u lies in V 2,α,[δ,T ]
...

To describe the regularity property of u(t) near a singular point, we take the
polar coordinates (ρ, θ) around a singular point such that
g = dρ2 + ρ2 (β + 1)2 dθ2 ,
˜
which is the standard cone metric of order β
...
We prove
˜
Theorem 1
...
Let u(t) be the solution given in Theorem 1
...
For any t > 0 and
q > 0 fixed, we have
˜
u(t) =
av v + O(q)
v∈T q

in a neighborhood of the singular point for some real numbers av , where
Tq =

k

k

ρ2j+ β+1 cos lθ, ρ2j+ β+1 sin lθ | l, j, k,

k−l
k
∈ N ∪ {0} ; 2j +
2
β+1

4

HAO YIN

˜
and O(q) is some error term satisfying that for any k1 , k2 ∈ N ∪ {0},
k ˜
(ρ∂ρ )k1 ∂θ 2 O(q) ≤ C(k1 , k2 )ρq

in the same neighborhood as above
...
1 of [14], an asymptotic expansion of the metric g(t) is given in
the form


Nj,k

ajkl (y)rj+k/β (log r)l  |z|

g∼

2β−2

2

|dz|
...

Theorem 1
...
In particular, it is proved that there is no log term
involved
...
3 also puts restrictions on the possible value of k and l
...
3, this refined information of expansion is
related to both the nature of the singularity and the structure of the equation
...
In particular,
we shall see log terms there
...
In Section 2, we define the function space in
which the weak solution lies, construct the weak solution via approximation and
prove various estimates for it
...
In Section 3, we show how a linear parabolic equation can
‘create’ regularity
...

The results in these two sections are somewhat independent from the application
...
The rest
three sections are devoted to the proof of three main theorems respectively
...
We assume without loss of generality that there is
only one singular point of order β, which is denoted by p
...
We
fix a background metric g , which is smooth away from the singular point and is the
˜
standard cone metric
g = (x2 + y 2 )β (dx2 + dy 2 )
˜
in the above mentioned coordinate neighborhood of p
...

By ‘polar coordinates’, we mean (ρ, θ) defined by
x = r cos θ,
and

y = r sin θ

1
rβ+1
...
Through˜
out this paper, we write B∗ for the subset {(ρ, θ) | ρ < ∗} and in the case that ∗ is
1, we simply write B
...
4
...
The proof of the main theorem in [24] contains a gap and therefore
the local existence result claimed there (with a rather weak initial data) is not proved
...
However, there are other problems in that paper
...
Recently, another gap was found and the
attempt to fix it directly motivates the definition of weak solution used in this paper
...
4 for details
...
It turns out that many seemingly important technical problems in [25] disappear naturally in this new exposition
...


Acknowledgment
...
He would like to thank the Mathematics Institute for
the wonderful working environment and Professor Topping for making it available
to him
...
Weak solution of linear equation
In this section, we develop a weak solution theory of linear equation of the type
∂t u = a(x, t) ˜ u + b(x, t)u + f (x, t)
...
1 with the definition of spaces of functions in which
the weak solution lies
...
More precisely, it is the usual C k,α space away from the
singularity and equipped with a H¨lder norm weighted naturally by the distance to
o
the singular point, while near the singularity, it is similar (a little stronger) to the
Sobolev space V21,0 used in the book [10]
...

In Section 2
...
2
...
4, we prove a
maximum principle, which extend the estimates in Theorem 2
...
Finally, in Section 2
...
This estimate is to be used
in the proof of local existence of the Ricci flow
...
1
...
We start by recalling some weighted H¨lder space defined
o
in [24]
...
In this paper, for simplicity, we denote them by E l,α and P l,α,[0,T ] respectively
...
We define the E l,α norm to be
f

E l,α

=

sup
k=0
...


6

HAO YIN

Here Br is {(ρ, θ) ∈ B | ρ < r}, C l,α (B1 \B1/2 ) and C l,α (U ) are just the usual H¨lder
o
norms
...
∞

+ f

C l,α (U ×[0,T ])


...

In spite of the tedious definition, it is not difficult to understand the meaning
of these weighted H¨lder space
...
Near a singularity, the E l,α norm is the bound for up to l−th
o
derivatives which one may obtain for a bounded harmonic function via applying
the interior estimate on a ball away from the singularity
...

These spaces are too weak for a useful discussion of the Ricci flow equation
because they contain almost no information at the singular point
...
For timeindependent functions, we define W l,α to be the set of function u in E l,α satisfying
1/2

2

|u|W :=

˜ u dV
˜

< +∞
...


For functions defined on S × [0, T ], we define V l,α,[0,T ] to be the set of function
u in P l,α,[0,T ] satisfying
1/2

|u|V [0,T ] := max |u(t)|W +
t∈[0,T ]

2

˜
|∂t u| dtdV

< +∞
...


For later reference, we need the following variations of W l,α and V l,α,[0,T ] :
• If Ω is a domain in S containing p, or a part of the infinite cone R+ × S 1
with standard cone metric g = dρ2 + ρ2 (β + 1)2 dθ2 containing the cone tip,
˜
l,α
we can define W (Ω) and V l,α,[0,T ] (Ω) similarly
...
Similar convention holds for [t1 , t2 )
and (t1 , t2 ]
...
1
...
If it satisfies some linear equation or the Ricci flow
equation classically away from the singularity and |u|V [0,T ] is finite, then we call it
a ‘weak’ solution
...

The next two lemmas show the advantage of having |u|W and |u|V [0,T ] bounded
respectively
...
2
...
If ˜ u is integrable
on S, or v · ˜ u is bounded from below (or above) by some integrable function, then
˜u = 0

v · ˜u = −

and

S

S

˜ u · ˜ v
...
Since the first inequality follows from the second one by letting v ≡ 1, we
prove the second one only
...

0

∂Bρ

Recall that here Bρ is the ball of radius ρ centered at the singularity measured with
respect to the cone metric g
...

(2
...

0
For each ρi , the integration by parts gives
˜
v · ˜ u + ˜ u · ˜ vdV =

(2
...

By (2
...


∂Bρi

∂Bρi

Together with (2
...


lim

i→∞

S\Bρi

The assumptions of the lemma imply that the integrand v · ˜ u+ ˜ u· ˜ v is bounded
from below (or above) by some integrable function, which allows us to conclude that
˜
v · ˜ u + ˜ u · ˜ vdV = 0
...
3
...


(2
...
4)

max |ϕ| + |∂t ϕ| < +∞,

S×[0,T ]

8

HAO YIN

S

˜
Ψ(u)ϕdV is an absolutely continuous function of t and

(2
...
In particular, the result holds for u ∈ V 2,α,[0,T ]
...
For any 0 ≤ t1 < t2 ≤ T , the Fubini theorem implies that
t2
t1

˜
Ψ (u)∂t uϕ + Ψ(u)∂t ϕdV dt
S
t2

=
S

˜
∂t (Ψ(u)ϕ(x, t))dt dV

t1
t2

˜
Ψ(u)ϕ(x, t)dV

=
S

,
t1

where the first line is absolutely integrable by (2
...
4) and the boundedness of
˜
u
...
5) holds
...
4
...
By Lemma 2
...
Since this quantity appears naturally in the
energy estimate of linear parabolic equations, we add it to the definition of weak
solution
...
2
...
In this section, M is a compact
surface with nonempty boundary and a Riemannian metric g
...
We prove two apriori estimates for the C 2,α solutions of
˜
linear parabolic equations on M
...
1
...
In the next section, we consider a sequence of surfaces with boundary
which approximates the conical surfaces and the geometry of this sequence is not
uniform
...
6)
u(0) = u0
on M

∂ν u = 0
on ∂M
...

(2
...


Proposition 2
...
For a, b, f in C α (M × [0, T ]) with 0 < λ < minM ×[0,T ] a and
u0 ∈ C 2,α (M ) satisfying (2
...
16) such that for t ∈ [0, T ],
t

(2
...


and

2

˜ u (t)dV +
˜

(2
...
The existence and uniqueness of the solution u in C 2,α (M × [0, T ]) is well
known (see Theorem 5
...
For (2
...
8) is the solution of the ODE
dh
= C1 h + C2
dt

(2
...


The proof of (2
...
Subtracting (2
...

By the definition of C1 and C2 and the fact that h ≥ 0, we have
∂t (u − h) ≤ a(x, t) ˜ (u − h) + b(x, t)(u − h)
...
Since u0 ≤ h(0), the classical maximum principle for
the linear parabolic equation on manifolds with boundary gives u ≤ h on M ×[0, T ]
...
For the proof of (2
...
11)
|∂t u| a−1 dsdV ≤
∂t u ˜ udsdV + Ct
...
12)
M ×[0,t]
t

˜ ∂t u · ˜ udV ds
˜

= −
0

= −
M

=

1
2

M

M
t

2
1
˜
∂t ˜ u dsV
0 2
2
˜ u (0)dV − 1
˜
2

2

˜ u (t)dV
...
The computation above involves some higher derivative of u
which does not exist for a function in C 2,α
...
Therefore, the computation can be justified
by smooth approximations
...
9) follows from (2
...
12)

10

HAO YIN

2
...
Weak solution of the linear equation via approximations
...
The idea is to approximate (S, g ) by a sequence
˜
of surfaces with boundary, solve a sequence of linear parabolic equations with Neumann boundary condition and take the limit of the sequence of solutions
...
6
...
This is different from [25] where the same type of approximation
was used again and again
...
e
...

Theorem 2
...
For a, b, f in V 0,α,[0,T ] with 0 < λ < minS×[0,T ] a and u0 ∈ W 2,α ,
there exists a weak solution u ∈ V 2,α,[0,T ] to
(2
...


Moreover,

t

(2
...
15)

|u|V [0,T ] ≤ C3

where C3 depends on the a
of a, b, f, u and λ
...
Define
Sk := S \ (ρ, θ)| ρ <

1
k


...

˜
˜
To use the linear estimate for surfaces with boundary in Section 2
...
7)
...

By the definition of Sk , we know that (ρ, θ) parametrizes a neighborhood of ∂Sk
1
for ρ ∈ [ k , 1) and θ ∈ S 1
...

Lemma 2
...
For each k fixed, there is so small that if u0,k : Sk → R is obtained
from u0 by a modification near ∂Sk given by
u0,k (ρ, θ) = u0 (η (ρ), θ),

ANALYSIS ASPECTS OF RICCI FLOW ON CONICAL SURFACES

11

then
u0,k

C 0 (Sk )

≤ u0

C 0 (S)

and
˜ u0,k
Moreover, we may take

L2 (Sk ,˜)
g

≤ 2 ˜ u0

L2 (S,˜)
g


...


Proof
...
For the second inequality, we assume
˜ u0
> 0, otherwise there is nothing to prove
...


Notice that ˜ (u0,k − u0 ) is supported in
(ρ, θ) |

1
1
≤ρ≤ +2
k
k

and ˜ u0 , hence ˜ u0,k is bounded (by a constant depending on u0 and k)
...


Now we can define an initial-boundary value problem

on Sk × [0, T ]
 ∂t u = a(x, t) ˜ u + b(x, t)u + f (x, t)
(2
...

Proposition 2
...
16) satisfying
t

(2
...
18)

˜ uk
Sk

2

≤ eC 1 t

u0

and C2 = f
t

C 0 (S)

e−C1 s C2 ds ,

+
0

C 0 (Sk ×[0,T ]) ,

and
2

2

˜
(t)dV +

˜
|∂t uk | dsdV ≤ C3
0

Sk

˜ u0 dV + C4 t,
˜
Sk

where C3 depends on a C 0 (Sk ×[0,T ]) and C4 depends on the C 0 norm of a, b, f, uk
and λ
...
17) and the Schauder estimate, we obtain some uniform estimate for
uk on Ω × [0, T ] (for k sufficiently large), for any fixed compact set Ω ⊂ S \ {p}
...
13) (defined on S × [0, T ]) and the convergence is in C 2,α on
Ω × [0, T ] for α ∈ (0, α)
...
17) and (2
...
14) and (2
...

Moreover, the P 2,α,T norm of u follows from the equation (2
...
This concludes the proof of Theorem 2
...


12

HAO YIN

2
...
Maximum principle and the uniqueness of weak solution
...
7 is unique
among all weak solutions if the time derivative of a is bounded
...
There are other
types of maximum principles in the literature using barrier functions
...
4 in [3]
...
2 there
...
9
...
19)

and

|b| , |∂t a| < λ−1

for some λ > 0
...
20)

max

˜u

L2 (S,˜)
g

t∈[0,T ]

˜
|∂t u| dV dt < ∞
...

Remark 2
...
For u ∈ V 2,α,[0,T ] , the assumption (2
...

Proof
...
3 implies that
˜
(u )2 a−1 dV is absolutely continuous so that we can compute
S +
d
dt

˜
(u+ )2 a−1 dV

˜
2u+ (∂t u)a−1 dV +

=

S

˜
(u+ )2 ∂t (a−1 )dV

S

S

˜
2u+ ˜ udV + C

≤

˜
(u+ )2 dV

S

S
2

≤

˜
˜ u+ dV + C

−2

˜
(u+ )2 a−1 dV
...
2 to justify the integration by
˜
parts
...

As a corollary of Lemma 2
...

1 A little more effort is necessary here because u is not in W 2,α
...
6 of [5]
sequence of nonnegative ui ∈

˜ u+

2

In fact, by setting ρ =
W 2,α ) satisfies

i→∞ S
es , we can

sup u+ (s − k, θ)
k∈N

˜
dV is finite, which implies that there exists a

such that ui converges to u+ in the sense that
˜ (ui − u+ )

lim sup |ui − u+ | +

(2
...
22)

W 2,α

S

2

˜
dV = 0
...


The standard mollification on the infinite cylinder (−∞, 0] × S 1 gives nonnegative ui , which
approximates u+ in the C 0 norm and the L2 norm of gradient
...
21) if we regard ui
as a function of (ρ, θ) (by the conformal invariance of the Dirichlet energy)
...
22)
...
20) and the inequality satisfied by u,
˜ u is bounded from below by some integrable function for almost every t
...
11
...
23)
∂t u = a(x, t) ˜ u + b(x, t)u + f (x, t),
whose coefficients satisfy (2
...
If u satisfies (2
...


Proof
...
5
...
24)
dt
and notice that h(t) is exactly the right hand side of the inequality we want to
prove
...
24) from (2
...

Applying Lemma 2
...
The
other side of the inequality can be proved similarly
...
Once we
know a weak solution is ‘the’ weak solution, then it coincides with the solution
given by Theorem 2
...

Theorem 2
...
Suppose that u is a weak solution on S × [0, T ] to the equation
(2
...
Assume that a, b, f are in V 0,α,[0,T ] and
max |∂t a| < ∞
...
26)

S×[0,T ]

Then u is the same as the solution given in Theorem 2
...
In particular, (2
...
15) holds for u
...

We use it to obtain C 0 and |·|V [0,T ] estimates for weak solutions as long as (2
...

Proof
...
25) given by Theorem 2
...
Then u − u is
˜
˜
a weak solution to the homogeneous equation
∂t (u − u) = a(x, t) ˜ (u − u) + b(x, t)(u − u)
˜
˜
˜
with (u − u)(0) = 0
...
9 to both u − u and u − u proves the
˜
˜
˜
theorem
...
2 to get
˜
2ui ˜ u + 2 ˜ ui · ˜ udV = 0
...

S

Here we used the Fatou’s lemma for the sequence ui ˜ u, which is uniformly bounded from below
by some integrable function
...

Lemma 2
...
Suppose a, b, f, u0 satisfy the same assumption as in Theorem 2
...


(2
...
13), then for any p > 1, we have
p

˜
|u| (t)dV ≤ C,

∀t ∈ [0, T ]

S

where C depends on p, T , the C 0 norm of a, a−1 , b, f, ∂t a and

p

S

˜
|u0 | dV
...
We will split u into three parts, u+ , u− and uf and estimate the Lp norm
of them separately
...
7
...

S

We can obtain u± by splitting u0 into its positive and negative parts, adding a
0
small positive constant and smoothing them out if necessary
...
7 gives us
u± (t) satisfying
∂t u± (t) = a ˜ u± + bu±
with u± (0) = u±
...
12, we know u = u+ − u− + uf
...
Since the
proof is the same, we give only the proof of u+
...
9
that u+ (t) is nonnegative
...


(2
...
29)
S×[0,T ]

which justifies the following computation
d
dt

˜
(u+ )p a−1 dV

˜
∂t (u+ )p a−1 + (u+ )p ∂t (a−1 )dV

=

S

S

˜ (u+ )p dV + C
˜

≤
S

˜
(u+ )p a−1 dV
S

˜
(u+ )p a−1 dV
...
2 to see that S ˜ (u+ )p dV vanishes
...

2
...
Estimates of the time derivative of solution
...
This is usually proved by taking derivative
of the equation
...
However, we can still take the time derivative
...
30)
∂t u = a(x, t) ˜ u + f (x, t)
and write w for ∂t u, we obtain
(2
...

a

Theorem 2
...
In addition to the assumptions that a, f ∈ V 0,α,[0,T ] , u0 ∈ W 2,α
and a−1 C 0 (S×[0,T ]) ≤ C, which is assumed in Theorem 2
...
31)
...
31) and ∂t u V 2,α,[0,T ] is bounded by a constant depending on the V 0,α,[0,T ] norm
of a, f, ∂t a, ∂t f , the W 2,α norm of u0 and w0 and a−1 C 0 (S×[0,T ])
...
We start by observing that all the rest conclusions in Theorem 2
...
31)
...
12 applies directly to show ∂t u is the unique weak solution
so that we have the bound of ∂t u V 2,α,[0,T ] as given in Theorem 2
...

To see the claim holds, let w be the weak solution to (2
...
7) with the initial data w0 and set
t

(2
...

˜
0

Theorem 2
...
31)
...
30), because u(t) and u(t) will then be two weak solutions with the
˜
same initial data so that Theorem 2
...
Here we used the fact that
∂t a is bounded
...
30)
...
30) pointwisely away from the singularity
...
33)
∂t u|t=0 = w(0) = a(x, 0) ˜ u0 + f (x, 0) = a(x, 0) ˜ u0 + f (x, 0)
...
28) and (2
...


16

HAO YIN

and compute
=

∂t w − ∂t a ˜ u − a ˜ w − ∂t f
˜
˜
˜

=

∂t w − ∂t aa−1 (w − f − H) − a ˜ w − ∂t f
˜
˜
˜

=

∂t H

∂t aa−1 H
...
Now, since H(0) = 0 by
˜
(2
...

To see that u is a weak solution, we notice first that
˜
T

T

2

˜
w2 dV ds < ∞,
˜

˜
|∂t u| dV ds =
˜

(2
...
For the Dirichlet energy bound of u, we take any compact
˜
˜
domain W in S away from the singular point and use the dominated convergence
theorem and the fact that ˜ w is bounded on W × [0, t] to get
˜
t

˜ u = ˜ u0 +
˜

˜w
˜

in W
...

˜
˜

˜
˜ u0 dV + 2t
0

S

S

t

2

2

˜
˜ u (t)dV ≤ 2
˜

(2
...
35) and (2
...
30) and hence
˜
concludes the proof of the theorem
...
Smoothing effect of linear equation
The estimates proved in previous section are as good as the initial data
...

However, as is well known for the linear parabolic equation on smooth manifolds,
rough initial data can be smoothed out
...
In this section,
we collect a few results in this direction for the linear parabolic equation on conical
surfaces
...

3
...
H¨lder regularity for bounded solution
...
Recall that we have a background metric g on S, hence, for
˜
any x, y ∈ S (including the singular point), we have a well defined distance function
˜
d(x, y), which is the infimum of the lengths of all smooth paths connecting x and y
...
Similarly, we have a parabolic version for functions
defined on S × [0, T ] for some T > 0
...


Remark 3
...
The definition above depends on g
...
Since it is a local result,
o
we state it in a neighborhood of the singular point p
...
In fact, the proof given below is a modification
of the proof in the smooth case which is well known
...
2
...
1)

∂t u = a(x, t) ˜ u + b(x, t)u + f (x, t)
...

It is closely related to the well known H¨lder regularity result for linear parabolic
o
equations of divergence form with bounded coefficients, for example, Section III
...
The idea is that in some
natural coordinates, the linear equation here can be shown to have bounded (but
not continuous) coefficients
...
In the parabolic case, we
have some extra difficulty caused by the fact that (3
...

This is why we assume ∂t a C 0 (S×[0,T ]) is finite
...
2, by using some natural coordinates, we first transform Theorem 3
...
3 below which has nothing to do with the conical
surface
...

1
2
1
2
1
2
ρ2

The observation is that these coefficients are bounded (not continuous at (0, 0))
...
1) in coordinates
˜
(x1 , x2 ) as
(3
...

g

18

HAO YIN

Here g = detgij
...
3)

t∈[0,T ]

T

2

2

|∂i u| dx +

max
B

|∂t u| dxdt < +∞
...
In summary, to show Theorem 3
...
3
...
2) satisfying (3
...
Assume that the coefficients of (3
...


If u is bounded on B \ {0} × [0, T ], then for each σ > 0, there is α > 0 depending
only on λ and C2 depending on σ, λ, T and u C 0 (B\{0}×[0,T ]) such that
u

C α (B1−σ ×[σ,T ])

≤ C2
...
1 in Chapter III of [10] except that
the principal part of (3
...
We will show that this is
not a problem if we assume ∂t g is bounded as above
...
3)) as if the equation is of the divergence form
...
Given this observation, some routine computation shows that the proof
in [10] still works
...
2
...

Lemma 3
...
Let R+ × S 1 be the infinite cone with metric g = dρ2 + (β + 1)2 ρ2 dθ2
...
4)

∂t u =

gu
˜

on (R+ × S 1 ) × (−∞, 0)
...

Proof
...

un satisfies (3
...
Theorem 3
...
This is impossible for large n unless u is
a constant
...


ANALYSIS ASPECTS OF RICCI FLOW ON CONICAL SURFACES

19

The application of Theorem 3
...

In particular, in (1
...
Fortunately, we have
Theorem 3
...
Suppose that u ∈ V 2,α,[0,T ] (B) is a weak solution to
(3
...


Then for any σ > 0, there is some α > 0 depending on C1 and C2 depending on
C1 and σ such that
u C α (B1−σ ×[σ,T ]) ≤ C2
...
Setting v = e2u , (3
...

Taking (ρ, θ) coordinates, and setting (x1 , x2 ) as before, we can rewrite the above
equation as
1
1
˜
∂t v = √ ∂i g ij g ∂j v + f
...
2) with an additional 1/v
...
It is then evident from here that Theorem 3
...
34
...
2
...
The main result of this section strengthens Theorem 2
...

Theorem 3
...
Suppose that
(1) the V 0,α,[0,T ] norms of a, b, f, ∂t a, ∂t b and ∂t f are bounded by C1 ;
(2) u0 W 2,α ≤ C2 ;
(3) a > λ > 0 on S × [0, T ]
...
6)
∂t u = a(x, t) ˜ u + b(x, t)u + f (x, t)
with u(0) = u0 given by Theorem 2
...
Then for each t0 > 0, w = ∂t u is a weak
solution to
(3
...


The proof consists of three steps
...
Note that for each lemma, we assume the same assumptions as in Theorem
3
...
The first step is
4In fact, the actual structure of g ij √g in (3
...
What we need is only the

fact that this matrix is comparable with δij
...
7
...


Moreover, by the H¨lder inequality, we may always assume that 1 < q < 2
...
7) with initial data ∂t u(t1 ), which
lies in V 2,α,[t1 +δ,T ] for all δ > 0
...
However, we
shall have a growth estimate of it in terms of t − t1
...
8
...
7) such that
˜
(1) for any δ > 0,
w
˜

V 2,α,[t1 +δ,T ]

≤ C(C1 , C2 , λ, T, t1 , δ)
...
7,
w(t)
˜

C 0 (S)

˜
+ ˜ w(t)

L2 (S,˜)
g

≤

C(C1 , C2 , λ, t0 )

...
6
...
6 is
Lemma 3
...
w constructed in Lemma 3
...

˜
So the proof of Theorem 3
...
7, Lemma 3
...
9
...
7
...
6)
...
7, it suffices to consider the growth of
˜ u(t1 ) (near the singular point) in B
...
2, we know u(t1 ) is C α (B)
for some α > 0, which means that in terms of polar coordinates
(3
...


For any y = (ρ0 , θ0 ) (ρ0 = 0), there is constant σ depending only on β such that
the geodesic ball Bσρ0 (y) (with respect to g ) is diffeomorphic to a disk embedded in
˜
S \ {p} and g restricted to Bσρ0 (y) is flat
...
Set
v(z1 , z2 , t) = u(σρ0 z1 , σρ0 z2 , (σρ0 )2 t + t1 )
...


By the interior estimates of linear parabolic equations, we know that
(3
...
(3
...

0

(3
...
9) and (3
...
2 proved in the appendix
...

0
Therefore, there is some q > 1 such that ˜ u(t1 ) is in Lq (S, g )
...
8
...
By the linearity, it
suffices to solve the following two problems separately and add up the solutions:
˜
∂t w1 = a ˜ w1 + ˜ 1 + f
bw
on S × [t1 , T ]
(3
...


(3
...

˜
b
Note that by the assumptions of Theorem 3
...
By
b, ˜
Theorem 2
...
11) satisfying
w1

V 2,α,[t1 ,T ]

≤ C(C1 , C2 , λ, T ),

so that to show Lemma 3
...
12) which satisfies
(1) and (2) in Lemma 3
...

The most natural way of solving (3
...
For that purpose, let w2,n,0 be a sequence of W 2,α functions
˜
such that
lim w2,n,0 − ∂t u(t1 ) Lq (S,˜) = 0
g
n→∞

and for each compact domain W ⊂ S \ {p}
(3
...


Note that in the last line above, we used the fact that ∂t u satisfies (3
...
6
...
7 gives a weak
solution w2,n (t) defined on S × [t1 , T ] satisfying
∂t w2,n = a ˜ w2,n + ˜ 2,n and w2,n (t0 ) = w2,n,0
...
14)

max
t∈[t1 ,T ]

w2,n (t)

Lq (S,˜)
g

≤ C;

(b) For any compact domain W ⊂ S \ {p}, there is C > 0 such that
w2,n

C 2,α (W ×[t1 ,T ])

≤ C;

(c) For any t ∈ (t1 , T ],
(3
...


22

HAO YIN

Before we start proving (a)-(d) above, we show how they imply Lemma 3
...
Recall
that our aim is to find w2 solving (3
...
8
holds for w2 in the place of w
...
13) imply that w2,n subconverges to a solution of
(3
...
By taking the limit, the estimates (c) and (d) become (2) and (1) for w2 in
Lemma 3
...

Now, let’s turn to the proof of the claim, i
...
(a)-(d)
...
13, (b) follows from (a) by applying known parabolic
¯
interior estimates to w2,n on Q × [t1 , T ], where W ⊂ Q ⊂ Q ⊂ S \ {p} and (d)
follows from (c), by regarding w2,n (t) as a weak solution to (3
...
12
...
For some t fixed, there
is a sequence of points p1 , · · · , pN in S, where p1 = p is the singular point such
that the following is true
...
Here cβ ∈ (0, 1) is a
constant depending only on β and the geometry of (S, g ) so that each Bi (i > 1)
˜
λ
is topologically a ball and p ∈ Bi for i > 1
...
We require that
1/4
¯
Bi cover S and that each point in S is covered by Bi for at most c times for a
universal constant c
...
10
...

Assume
max |∂t a| < C1

Bi ×[0,t]

for some C1
...
16)

|u| ≤ C2

max
1/2

Bi

×[t/2,t]

1/q

t

1

q

˜
|u| dV dt

t2/q

0

Bi

and
(3
...
16) of this lemma is a Moser iteration, which is more or less standard, while the proof of (3
...
16) with Theorem 3
...
In
order not to distract the readers from the proof of Theorem 3
...

We apply Lemma 3
...
16) and (a) that
w2,n (t)

C 0 (S)

≤

C3
,
(t − t1 )1/q

ANALYSIS ASPECTS OF RICCI FLOW ON CONICAL SURFACES

23

which is the C 0 part of (3
...
For the other half,
N

2

˜ w2,n (t) dV
˜

2

≤

S

i=1

1/4
Bi

N

≤

2
C2
i=1

≤

˜ w2,n (t) dV
˜
q

˜
|w2,n | dV dt
t1

Bi
2/q

t

C4
(t − t1 )4/q

2/q

t

1
(t − t1 )4/q

q

˜
|w2,n (t)| dV dt
t1


...
7), then for x, y ∈ [0, ∞), we have x2/q +y 2/q ≤ (x+y)2/q
...
8
...
9
...
9
...

˜
t1

Obviously, u(t1 ) = u(t1 )
...
First, we show that
˜
u − u satisfies some homogeneous equation with an error term, i
...
(3
...

˜
Then we show that although u − u is not a weak solution in V 2,α,[t1 ,T ] , there is still
˜
some control over its energy and time derivative when t is close to t1 , so that in the
final step, we can invoke an argument similar to the proof of Lemma 2
...

˜
Step 1
...

˜
˜
˜
u
For t > t1 , we set
H = ∂t u − a ˜ u − b˜ − f = w − a ˜ u − b˜ − f
˜
˜
u
˜
˜
u

(3
...

u
Here in the last line above, we used the fact that w satisfies (3
...
For each fixed
˜
x ∈ S \ {p}, consider H as a function of t alone satisfying the above ODE with
initial data H(0) = 0
...
19)

|H| (t) ≤ C6

|˜ − u| ds
...
18) and (3
...

u
u
u

(3
...
19), (3
...
However, there is some technical issue in the maximum
˜
principle type argument, which requires the following claim
...
We claim that u satisfies
˜
˜ u(t)
˜

(3
...

˜ ˜

(3
...
22) follows from (2) of Lemma 3
...
To see (3
...
By (2) in Lemma 3
...
19)
...
20) on W × [t1 , T ] to see that ˜ u is bounded on W × [t1 , T ]
...
This together with
(2) in Lemma 3
...

˜
˜ ˜
t1

W
2

W

˜ u dV is absolutely continuous function on
˜
˜

=

The Fubini theorem implies that

2

[t1 , T ] and
d
dt

2

˜
˜ u dV
˜

˜ u · ˜ wdV
˜
˜ ˜
W

W

≤ 2 ˜u
˜

C
(t − t1 )1/q

≤
Since we know

˜ u(t1 )
˜

≤

L2 (W,˜)
g

L2 (W,˜)
g

˜ u(t1 )

˜u
˜

L2 (S,˜)
g

˜w
˜

L2 (W,˜)
g

L2 (W,˜)
g


...
By the arbitrariness of W , (3
...

˜
Step 3
...
(3
...
3 imply that F is
u
an absolutely continuous function of t so that we can compute
d
F (t) ≤ 2
dt

˜
(˜ − u)( ˜ (˜ − u) + a−1 b(˜ − u) + a−1 H)dV + CF (t)
u
u
u
S

˜
H 2 dV
...
2,
(3
...
18), (3
...
22)
...

t1

In summary, we have
(3
...
One can conclude from the above inequality that F ≡ 0
...

s∈[t1 ,t]

˜
It turns out that F is also absolutely continuous and
d ˜
d
F (t) ≤ max 0, F (t)
dt
dt

t

≤ max 0, CF (t) + C

F (s)ds

˜
≤ C F (t)
...
9
...
Global existence of conical Ricci flow
In this section, we prove Theorem 1
...
The local existence is proved in Section 4
...
As a result of the local existence result (Theorem
4
...
8)
...
2 and Section 4
...
1
...
1
...
In this section, we prove the local existence of the Ricci flow equation
r
˜
(4
...

2
Here r is some constant
...

Theorem 4
...
Suppose that u0 : S → R and the curvature K0 of e2u0 g are both
˜
in W 2,α
...
1)
...

The proof is an iteration
...
Let ui (i > 1)
to be the weak solution (given by Theorem 2
...
2)

∂t ui = e−2ui−1 ˜ ui +
ui (0) = u0
...


26

HAO YIN

Lemma 4
...
For ui defined above, if we write wi (i ≥ 1) for ∂t ui , then wi (i ≥ 2)
is the weak solution of
r
∂t wi = e−2ui−1 ˜ wi − 2wi−1 (wi − 2 )
r
−2u0 ˜
−2u0 ˜
wi (0) = e
u0 + 2 − e
K
...
3)

Proof
...
For i = 2, we try to apply Theorem 2
...

2
2

Since ∂t a = ∂t f = 0, it remains to check w0 ∈ W 2,α , which follows from the
assumption that K0 is in W 2,α
...
14 proves that w1 is a weak solution to
(4
...

Assume the lemma is proved for i − 1
...
w0 remains as before, so that we can apply
Theorem 2
...

To show that ui converges to the solution we want, we need several apriori
estimates
...
Set
˜
C = max

u0

C 0 (S)

, e−2u0 ˜ u0 +

r
˜
− e−2u0 K
2

C 0 (S)


...
3
...

Proof
...
Assume that
it is true for i − 1
...
2) and (4
...
2) and (4
...
12 to see that
t

(4
...
5)

wi (t)

C 0 (S)

≤ eC3 t

e−2u0 ˜ u0 +

r
˜
− e−2u0 K
2

t
C 0 (S)

e−C3 s C4 ds
...
12, we have
C1 = 0
C3 = −2wi−1

r
˜
− e−2ui−1 K
2
C4 = rwi−1 C 0 (S)

C2 =
C 0 (S)

C 0 (S)

By induction hypothesis again, we can bound C1 , · · · , C4 by a number depending
˜
only on C
...
4) and (4
...

The next lemma provides C 2,α estimates of ui and wi away from the singularity
...
4
...
3, we have
ui

P 2,α,[0,T ]

, wi

P 2,α,[0,T ]

≤C

˜
for some constant C depending on C and the W 2,α norm of u0 and e−2u0 ˜ u0 +
r
−2u0 ˜
K
...
Note that this lemma is not proved by induction
...

By the boundedness of wi and ui and (4
...
6)

C 0 (S)

˜
≤ C(C)

for all t ∈ [0, T ]
...
7)

max ui (t)

t∈[0,T ]

C α (S)

˜
< C(C) < ∞

for some α ∈ (0, 1) depending only on β
...
Let (x, y) be the conformal coordinates defined in B, then
˜ ui (t) = (x2 + y 2 )−β ui ,
which implies (by (4
...
By the Lq estimate of
and the Sobolev embedding theorem, there is a
function v in C α (B) ∩ W 1,2 (B) with
v = (x2 + y 2 )β ˜ ui (t)

on

B

and
v|∂B = ui (t)
...

This proves the claim
...
(4
...
2) to get a uniform
bound of ui P 2,α ,[0,T ]
...
2) to get the uniform bound
ui P 2,α,[0,T ] as we need
...
3)
...
4, ui subconverges
...

Lemma 4
...
There exists T > 0 (maybe smaller than given in Lemma 4
...

Proof
...
2) that
∂
˜
(ui+1 − ui ) = e−2ui ˜ (ui+1 − ui ) + (e−2ui − e−2ui−1 ) ˜ ui − (e−2ui−1 − e−2ui )K
...
Since
˜ ui by (4
...


Moreover, ∂t a is bounded so that Theorem 2
...


Hence, if we choose T small, then the sequence ui is Cauchy in C 0 norm and the
lemma is proved
...
By Lemma 4
...
1) pointwisely
...
1, we still need
Lemma 4
...
For T determined above,
|ui |V [0,T ] , |wi |V [0,T ] ≤ C(T )
...

Proof
...
2) and (4
...

Since ∂t (e−2ui−1 ) is uniformly bounded, we obtain control of |ui |V [0,T ] and |wi |V [0,T ]
by Theorem 2
...
Note that the constants C3 and C4 in (2
...
2) and (4
...
3
...
1, we can discuss the blow-up criterion which serve as a
starting point for the proof of long time existence
...

Lemma 4
...
For i = 1, 2, suppose that ui ∈ V 2,α,[0,T ] is a weak solution to Ricci
flow (1
...
If the Gauss curvature of e2ui g are bounded on S ×[0, T ]
˜
and that u1 (0) = u2 (0), then u1 = u2 on S × [0, T ]
...
We subtract the equation of ui to get
˜
∂t (u1 − u2 ) = e−2u1 ˜ (u1 − u2 ) + (e−2u1 − e−2u2 )( ˜ u2 − K),
where
1

e−2u1 − e−2u2 = −2(u1 − u2 )

e−2u2 −2t(u1 −u2 ) dt
...
9 to show u1 ≡ u2
...
1
...
Theorem 4
...
The next
lemma gives a characterization of Tmax
...
8
...
1 and Tmax
is defined as above
...
Moreover, if Tmax < +∞, then
(4
...

˜
In the rest of this paper, we shall call the solution given in the above lemma the
maximal solution starting from u0
...
By the definition of Tmax , for each T < Tmax , there is uT ∈ V 2,α,[0,T ]
solving (1
...
For any t < Tmax , we define
u(t) = uT (t) for any t < T < Tmax , which is well defined by Lemma 4
...
It remains
to show (4
...

If the lemma is not true, then there is C > 0 such that
|K| ≤ C
...
1) is equivalent to ∂t u =

r
2

− K, we know
|u| ≤ C

sup
S×[0,Tmax )

for possibly another C > 0
...
4, we have
(4
...

We claim that for all T < Tmax ,
(4
...


To see the claim, we regard u as the weak solution of the linear equation
∂t u = a(x, t) ˜ u + f (x, t)
r
˜
where a = e−2u and f = 2 − e−2u K
...
12
0
to bound |u|V [0,T ] by T , the C norm of a, f, u and a−1 and the W 2,α norm of u0
...

With (4
...
10), we can extend the definition of u to Tmax such that

u(Tmax )

W 2,α

+ ˜ u(Tmax )

W 2,α

< +∞
...
1 shows that we can extend the domain of u to Tmax + δ, while
keeping u V 2,α,[0,Tmax +δ ] and ∂t u V 2,α,[0,Tmax +δ ] finite
...

Finally, to conclude this section, we prove that for a natural choice of r, the
maximal solution of (1
...
Following [22], we set
χ(S, β) = χ(S) +

βi ,

where χ(S) is the Euler number of the underlying Riemann surface S
...
9
...
1 and
that u(t) is the maximal solution starting from u0
...
11)

2πχ(S, β) =

Kt dVt
...

˜
Proof
...

2πχ(S, β) =
S

˜
For any t ∈ [0, Tmax ), by Kt = e−2u (− ˜ u + K), we have
˜ ˜
− ˜ u + KdV
...
2 implies that
˜ udV = 0,
˜
S

which proves (4
...

Since |u|V [0,T ] is finite for any T < Tmax , Lemma 2
...
2 allow us to
compute
d
˜ ˜
V (t) =
2 ˜ u + re2u − 2KdV = rV (t) − rV0
...

4
...
Apriori estimate for the conformal factor
...
8 implies that if
Tmax for a maximal solution is finite, then the curvature K(t) blows up as t → Tmax
...

Lemma 4
...
Suppose u0 is some initial data satisfying the assumptions of Theorem 4
...
Let u(t) be the maximal solution given by Lemma 4
...

There exists C > 0 depending on Tmax and u0 such that
u

C 0 (S×[0,Tmax ))

≤ C
...
In the smooth
a
case, if ϕ(t) is the potential function in the sense that
˜ ϕ(t) = e2u(t) − Vt
˜
V
where Vt is the volume of g(t), then (up to some normalization) ∂t ϕ satisfies a
linear parabolic equation, from which we obtain immediately C 0 apriori estimate

ANALYSIS ASPECTS OF RICCI FLOW ON CONICAL SURFACES

31

of u
...
10 by showing that this argument
works for conical surfaces as well
...
11
...

S

Then up to a constant, there is a unique u ∈ W 4,α such that
˜ u = f
...
10 of [25]) and hence is
moved to the appendix
...
9, Lemma 4
...
12)
because
S

rV0 ˜
rV0
dV =
= 2πχ(S, β) =
˜
2
2V

˜ ˜
KdV
...
The existence of potential
function ϕ(t) is given in the next lemma
Lemma 4
...
Suppose that u(t) is a maximal solution to (1
...
12)
...
13)
and

˜ ϕ = e2u − V0
...


Proof
...
9 and Lemma 4
...
14)

˜ ϕ0 = e2u0 − V0
...
13) with ϕ(0) = ϕ0
...
13), we get
t

(4
...


ϕ(t) = ert ϕ(0) +
0

We claim that
(4
...
14)
...


Here in the above computation, we used (1
...
12) and (4
...

It remains to check that |∂t ϕ|V [0,T ] and ∂t ϕ C 0 (S×[0,T ]) are finite for any T <
Tmax
...
13), it suffices to show |ϕ|V [0,T ] and ϕ C 0 (S×[0,T ]) are finite because
h0 W 4,α < ∞ and |u|V [0,T ] < ∞ by the definition of the maximal solution and
T < Tmax
...
15)
2

˜
˜ ϕ dV < ∞

max
t∈[0,T ]

S

and
max ϕ(t)

C 0 (S)

t∈[0,T ]

<∞

by using the fact that ϕ0 , h0 are in W 4,α and u(t) is in V 2,α,[0,T ]
...
13)
shows that ϕ is bounded on S × [0, T ], which is stronger than
2

˜
|∂t ϕ| dV ds < +∞
...
Using (4
...


+ r∂t ϕ

With this equation and the fact that u ∈ P 2,α,[0,T ] and ∂t ϕ(0) = rϕ0 +2u(0)+2h0 ∈
W 2,α , the interior Schauder estimate implies that ∂t ϕ is in P 2,α,[0,T ]
...

The final step in the proof of Lemma 4
...
12 applies to ∂t ϕ as a weak solution to the above equation to give the
required apriori C 0 bound
...
3
...
In this section, we prove Theorem
1
...
Suppose u0 satisfies the assumption of Theorem 4
...
9
...
8
...
If otherwise, Lemma 4
...
17)

u

C 0 (S×[0,Tmax ))

≤ C1 < +∞
...
8,
which asserts that
lim sup K(t) C 0 (S) = +∞
...
17), this is equivalent to
lim sup ∂t u
t→Tmax

C 0 (S)

= +∞
...


i→∞

By modifying xi and ti if necessary, we may assume
(4
...

2 S
2 t∈[0,ti ] S

For any T < Tmax , we can apply Theorem 3
...
19)

u

C α (S×[δ,Tmax ))

≤ C
...
1), so we may assume that xi converges to the unique singular point p
...

We compare the speed of xi → p and λi → ∞ and distinguish three cases
...
In fact, this case never happens because we can
˜
apply the theory of quasilinear parabolic equation to (1
...

dg (xi , p)2
˜
Case two: 0 < dg (xi , p)2 λi < ∞
...

˜
Suppose xi = (ρi , θi )
...

Set
wi (ρ, θ, t) = u(

ρ
1/2
λi

, θ, ti +

t
),
λi

which satisfies
(4
...

∂t
λi 2
We can apply the Schauder estimate in a neighborhood of (ρ, θ, t) = (1, 0, 0) to see
that wi converges in C 2 to a limit w∞ with
(4
...

This is a contradiction to (4
...

Case three: dg (xi , x0 )2 λi = 0
...
20)
˜
1/2
holds
...
Taking t-derivative of the equation
satisfied by wi , we have
1 r
1
˜
˜
2∂t wi e−2wi K
...
18), the term
(−2∂t wi ) ∂t wi −

1
λi

r
˜
− e−2wi K
2

+

1
˜
2∂t wi e−2wi K
λi

is uniformly bounded on {(ρ, θ, t)| ρ < 2, t ∈ [−1, 0]}
...
8), we know ∂t wi is a
weak solution defined on {(ρ, θ, t)| ρ < 2, t ∈ [−1, 0]}
...
2 then implies the
existence of α ∈ (0, 1) and C1 > 0 (independent of i) such that
∂t wi (0)

C α ({(ρ,θ)| ρ<1})

≤ C1
...
13
...

The point is that Theorem 3
...

This together with (4
...
We can then obtain a contradiction as in Case two
...
Higher regularity of conical Ricci flow
In previous sections, we proved the global existence of a Ricci flow solution
...
In this
section, we show that
Lemma 5
...
Suppose u is the solution in Theorem 1
...
If for some C1 > 0 and
T > 1, we have
u V 2,α,[T −1,T ] + ∂t u V 2,α,[T −1,T ] ≤ C1 ,
then for any k > 1, there exists C2 (k) depending only on C1 (not on T ) such that
k
∂t u

V 2,α,[T −1/2,T ]

≤ C2 (k)
...
2
...

Before we start the proof, we note that since ∂t u =
k−1
bound ∂t K
...
1
...

Lemma 5
...
Let u be the solution in Lemma 5
...
For any 0 < δ < 1, we have
∂t K

V 2,α,[T −δ,T ]


for some C depending on C1 in Lemma 5
...

We study the evolution equation of K instead of u,
∂t K = e−2u ˜ K + K(2K − r)
...
1)

The proof is very similar to that of Theorem 3
...
For δ ∈ (0, 1) in Lemma 5
...
The same proof as in Lemma 3
...
2)

for some q > 1
...
Taking t-derivative of
(5
...

Using ∂t u = −K + r/2 and (5
...
3)

∂t w = e−2u ˜ w + w(6K − 2r) − K(2K − r)2
...
3) as a linear parabolic equation of w, while the coefficients are in
V 2,α,[T −1,T ] and ∂t e−2u lies also in V 2,α,[T −1,T ]
...
2), Lemma 3
...

˜
Moreover, Lemma 3
...
4)

w
˜

V 2,α,[t0 +η,T ]

≤ C(η)

for 0 < η < T − t0

and
(5
...

We define for t ∈ [t0 , T ]
t

˜
K(t) = K(t0 ) +

w(s)ds
...
3 is done if we can show that K ≡ K for any t ∈ [t1 , T ]
...
9
...
1) at t0
...


36

HAO YIN

=

˜
∂t w − e−2u ˜ w − w(4K − r) + e−2u ˜ K(2∂t u)
˜
˜ ˜ ˜

=

˜ ˜
(2∂t u)(−H) + (r − 2K) w − K(2K − r) + ∂t w − e2u ˜ w − w(4K − r)
˜
˜
˜ ˜ ˜

=

∂t H

˜
˜ ˜
(2∂t u)(−H) + 4w(K − K) + (2K − r) K(2K − r) − K(2K − r)
...
Due to (5
...


On the other hand, by (5
...

(t − t0 )1/q

In summary, we obtained
|∂t H| ≤ C |H| +

C
˜
K −K ,
(t − t0 )1/q

from which we get by integration (using H(t0 ) = 0)
t

|H| (t) ≤ C

(5
...
Set
˜
F (t) := sup K(t) − K(t)

and FH (t) := sup |H|
...
6) implies that
t

1
F (s)ds
...
1) and the definition of H, we have
FH (t) ≤ C

(5
...


In order to apply Lemma 2
...
7), we check that (2
...
In fact, as in
Step 2 of the proof of Lemma 3
...

t1

S

Lemma 2
...
8)

F (t) ≤ C1

t

s

t0

t0

FH (t)dt ≤ C1
t0

1
F (r)drds
...
8) that F ≡ 0 for t ∈ [t0 , T ]
...

F (t) ≤ C1 C2
(1 − 1/q)(2 − 1/q)

ANALYSIS ASPECTS OF RICCI FLOW ON CONICAL SURFACES

37

Plugging this back into (5
...

(1 − 1/q)(2 − 1/q)(3 − 2/q)(4 − 2/q)

Repeating this process gives F ≡ 0 and proves Lemma 5
...

5
...
Higher order regularity
...
For higher t-derivatives, we can apply Theorem
l
3
...

To be precise, we claim that for l ≥ 2
l
l
l
∂t (∂t u) = e−2u ˜ (∂t u) + Pl · ∂t u + Ql

(El )

l−1
where Pl and Ql are polynomials of ∂t u, · · · , ∂t u with constant coefficients
...
Assume the claim is true for l
...


Hence, the claim is proved if we take
Pl+1 = −2∂t u + Pl
and
l
l
Ql+1 = 2∂t u(Pl · ∂t u + Ql ) + (∂t Pl ) · ∂t u + ∂t Ql
...
1 by induction
...
6 directly to it, because u, ∂t u, ∂t u are in V 2,α,[T −δ,T ]
...


The proof for higher order derivatives is similar and omitted
...
Asymptotic expansion of the solution
The aim of this section is to prove Theorem 1
...
The proof is built on the
l
previous knowledge that ∂t u is bounded for all l = 0, 1, 2, · · ·
...


38

HAO YIN

6
...
Formal consideration
...
On one hand,
we need to include sufficiently many terms so that u(t) can be expanded as a series
of such terms
...

The consideration in this subsection is a little formal, but it shall be fully justified
when we prove Theorem 1
...

First, let’s recall the expansion of bounded harmonic functions defined on B\{0}
...
1
...
e
...
Then we have
∞
k

k

ak ρ β+1 cos kθ + bk ρ β+1 sin kθ

u(ρ, θ) = a0 +
k=1

for ρ ∈ (0, 1)
...

The proof is a well known argument of separation of variables and is omitted
...
Namely, we should consider linear
combinations of the terms in
k

k

Th = ρ β+1 cos kθ, ρ β+1 sin kθ | k = 0, 1, 2, · · ·


...

Next, we would like to include more terms so that some basic algebraic operations
are closed
...

2
We characterize Ta in the following lemma
...
2
...

Proof
...
Moreover, Ta contains Th trivially
...
To see this,
we compute
1
1
2
cos 2θ + 1
,
ρ β+1 cos θ · ρ β+1 cos θ = ρ β+1
2
2
2
which implies that ρ β+1 should be in Span(Ta )
...

Finally we define
T =

k

k

ρ2j+ β+1 cos lθ, ρ2j+ β+1 sin lθ | l, j, k = 0, 1, 2, · · · ;

k−l
∈ N ∪ {0}
...

β+1
β+1
The motivation behind the definition of T is explained in the next lemma
...
3
...

Proof
...
For (2), for each u = ρσ cos lθ
in T , we compute
˜ ρσ+2 cos lθ

=
=

By the definition of T , σ + 2 >
we may take

1
l2
2
)ρσ+2 cos lθ
(∂ρ + ∂ρ − 2
ρ
ρ (β + 1)2
l2
((σ + 2)2 −
)ρσ cos lθ
...

(β + 1)2
The computation works as well if we replace cos by sin
...

v = ((σ + 2)2 −

6
...
Finite expansion
...
For that purpose, we shall define a class of
˜
functions O(q) for any nonnegative real number q
...

˜
Definition 6
...
A function u defined in B1/2 \ {0} is said to be in O(q) for q ∈
[0, ∞) if and only if there are constants Ck for each k = 0, 1, 2, · · · such that
(6
...


Remark 6
...
We note that (6
...


To define an expansion up to order q > 0, we consider only the linear combination
of functions in
k
k
k−l
k
T q = ρ2j+ β+1 cos lθ, ρ2j+ β+1 sin lθ | l, j, k = 0, 1, 2, · · · ;
∈ N ∪ {0} ; 2j +
...

Definition 6
...
A function u is said to have an expansion up to order q if and
only if there is a set of real numbers av for each v ∈ T q such that
˜
av v + O(q)

u=
v∈T

on

B1/2 \ {0}
...
7
...


40

HAO YIN

l0
Proof
...
By Lemma 6
...

R :=
l≥l0
k
For any k1 and k2 in N ∪ {0}, we need to show that (ρ∂ρ )k1 ∂θ 2 R is bounded on
B1/2 \ {0}
...
2)

l
l
al ρ β+1 −q cos(k2 ) lθ + ˜l ρ β+1 −q sin(k2 ) lθ ,
˜
b

k k
(ρ∂ρ 1 ∂θ 2 )R :=
l≥l0

where the exact formula for al and ˜l is not important
...
2) is a continuous function of (ρ, θ) defined on [0, 1/2] × S 1
...
To see this, we
use Abel’s uniform convergence test and write for ρ0 = 1/2
l

l

˜ β+1
al ρ β+1 −q cos(k2 ) lθ = al ρ0
˜
l

−q

l

cos(k2 ) lθ · (ρ/ρ0 ) β+1 −q
...

Lemma 6
...
If f1 and f2 both have expansions up to order q, then so do f1 ± f2 ,
f1 · f2 and ef1
...
The claim holds trivially for f1 ± f2
...

f1
Instead of showing that e has the required expansion, we prove something a little
stronger
...
By changing F (·) to F (c + ·), we may assume without loss of
generality that the constant term in the expansion of f1 vanishes
...

Recall that we have the Taylor expansion formula with the integral remainder,
n

F (x) =
l=0

F (l) (0) l
1
x +
l!
n!

1

F (n+1) (tx)(1 − t)n dt xn+1 ,
0

1
in which we choose n so that (n + 1) min 2, β+1
so far, we know
n
F (l) (0)
(f1 )n
l!

> q
...
It remains to show
1

˜
F (n+1) (tf1 )(1 − t)n dt (f1 )n+1 ∈ O(q)
...
3)
0

1
If q < min 2, β+1 , then ξ must be zero because T q contains nothing but a
1
˜
constant function
...
If q ≥ min 2, β+1 , by our choice of n

ANALYSIS ASPECTS OF RICCI FLOW ON CONICAL SURFACES

41

n+1
1
˜
˜
and the fact that f1 ∈ O(min 2, β+1 ), we have f1
is in O(q)
...
3) is
reduced to
1

F (n+1) (tf1 )(1 − t)n dt

(6
...


0

By direct computation, one can check that (6
...

6
...
Expansion of Conical Ricci flow solution
...
1
...
2, for any t > 0, we know that
l
∂t u(t) ≤ Cl

for l ∈ N ∪ {0}
...

We consider the following family of claims
...

Here are two easy observations
...
1, we know that the claim
C 0 is true
...

Hence, to show that the claim C q holds for any q > 0, it suffices to justify C qi for a
sequence qi going to ∞
...

¯
Lemma 6
...
Suppose that w and f are smooth functions on B1 \ {0} and that w
is bounded and f has an expansion up to order q ≥ 0
...
When q =
any k ∈ N ∪ {0}, w has an expansion up to order q = 2 + q
...
3
...
8 implies that the right hand side of (El ) for
l
l = 0, 1, · · · has an expansion up to order q
...
9, ∂t u has an expansion
q
up to order q with q > q + 1 for all l
...
Theorem 1
...

The proof of Lemma 6
...
10
...

k
Here q = 2 + q if q = β+1 − 2 for any k ∈ N ∪ {0} and q can be any number
smaller than 2 + q if otherwise
...
9
...
6, there is ξf ∈ Span(T q ) such that
˜
f = ξf + O(q)
...
3 implies the existence of ξw ∈ Span(T q+2 ) with
˜ ξw = ξf
...
10 gives wo ∈ O(q ) with ˜ wo = fo
...
By Lemma 6
...
Since w = wh + ξw + wo , w has an
expansion up to order q for q given in Lemma 6
...

The rest of this section is devoted to the proof of Lemma 6
...

Let’s first do some formal computation to motivate the proof
...
Assume we have an expansion
∞

Al (ρ) cos(lθ) + Bl (ρ) sin(lθ),

wo =
l=0

which is convergent in some suitable sense such that
∞

˜ wo =

(Ll Al ) cos(lθ) + (Ll Bl ) sin(lθ)
l=0

where

l2
1
2
Ll := ∂ρ + ∂ρ − 2

...


fo =
l=0

˜
Notice that it follows from the theory of trigonometric series that fo is in O(q) if
and only if
(6
...

Lemma 6
...
Suppose a∗ : (0, 1] → R satisfies that
(6
...


If
(6
...
8)

Ll A∗ = a∗

ANALYSIS ASPECTS OF RICCI FLOW ON CONICAL SURFACES

43

and
(6
...


Remark 6
...
The dependence of C2 (·) on C1 (·) is linear in the sense that if we
multiply every one of C1 (·) by a positive constant λ, then C2 (·) is multiplied by the
same constant
...
e
...

l
Proof
...
First, we note that the solution of (6
...
We assume that the solution is of the form A∗ (ρ) =
h(ρ)ρc , then (6
...

Hence,
ρ

h (ρ) = ρ−2c−1 h (1) +

a∗ (t)tc+1 dt
...
We can choose it to be anything we want, since it
suffices for the proof of the lemma to give one solution
...


h (1) +
1

Hence,
ρ

h (ρ) = ρ−2c−1

a∗ (t)tc+1 dt
...
6), we get
|h (ρ)| ≤

(6
...

c+2+q

On the other hand, we have
ρ

A∗ (ρ) = ρc h(1) +

(6
...

1

Here h(1) is another constant at our disposal
...
7), we have two possible cases:
Case 1: c > 2 + q
...

Case 2: c < 2 + q
...

1

Hence,
C1 (0)
ρ−c+2+q ≤ C2 (0)ρ2+q
(c + 2 + q) |−c + 2 + q|
for the same C2 (0) as in Case 1
...
11)
(ρ∂ρ )A∗ (ρ) = cA∗ + ρc+1 h (ρ),
which has the correct order of decay by (6
...
Moreover, C2 (1) in (6
...

For k1 > 1, we rewrite the equation Ll A∗ = a∗ as
(ρ∂ρ )2 A∗ −

l2
A∗ = ρ2 a∗ ,
(β + 1)2

The estimate (6
...
6) directly,
while for the case k1 > 2, we take ρ∂ρ repeatedly on both sides of the above
equation
...
11 to both al and bl to get Al and Bl
...
5) implies
C(k1 , k2 ) q
ρ
...
11 and Remark 6
...

lk2
Similar arguments work for bl as well
...
12)

lk2 (ρ∂ρ )k1 Al (ρ) + lk2 (ρ∂ρ )k1 Bl (ρ) ≤ C (k1 , k2 )ρq
...
Moreover, the decay of wo , (ρ∂ρ )k1 ∂θ 2 wo also follows from
(6
...


Appendix A
...
3
Similar to the proof of Theorem 10
...
In fact, the authors wrapped
up the argument of Di Giorgi iteration in Theorem 7
...
Our strategy of proving Theorem 3
...
1 there
...
1 explicitly
and simplify the definition and statement a little since we are not interested in
equations of the most general form
...


ANALYSIS ASPECTS OF RICCI FLOW ON CONICAL SURFACES

45

A
...
The class of functions B2
...
Let QT = Ω × [0, T ] for some T > 0
...
Following [10], we define
1/2

u

2,Ω

u2 dx

=
Ω

and
u

QT

= sup

u(t)

0≤t≤T

2,Ω

+ ux

2,QT


...

Definition A
...
A function u : QT → R is said to be in the class B2 (QT , M, γ) if
it satisfies
(a) there is constant C > 0 such that
u

QT

≤C

2

and that Ω u dx is a continuous function of t;
(b) supQT |u| ≤ M ;
(c) for w(x, t) = ±u(x, t), we have
(A
...
2)

w(k)

2
Q((1−σ1 )ρ,(1−σ2 )τ )

≤ γ [(σ1 ρ)−2 + (σ2 τ )−1 ] w(k)

2

+ µ3/4 (k, ρ, τ ) ,

2,Q(ρ,τ )

Here in the above w(k) = max {w − k, 0}, σ1 , σ2 are any number in (0, 1),
(x0 , t0 ) ∈ QT , ρ, τ are any positive number such that Q(ρ, τ ) ⊂ QT and
t0 +τ

|Ak,ρ (t)| dt

µ(k, ρ, τ ) =
t0

where Ak,ρ (t) = Kρ ∩ {w(x, t) > k} and |Ak,ρ (t)| is the measure of Ak,ρ (t)
...
2
...

Here is a simplified version of Theorem 7
...

Theorem A
...
[Theorem 7
...

There are θ > 0 and α ∈ (0, 1) depending only on γ such that the following holds
...
Here c depends on
γ, ρ0 and M
...
1) and (A
...

A
...
Check the assumptions in Definition A
...
Let u be as in Theorem 3
...

It is easy to see that (a) and (b) in Definition A
...
3 to show Ω u2 dx is an absolutely continuous function of t
...
Moreover, we want to make sure that γ
there depends only on λ
...
3), we apply Lemma 2
...
3)

B

√
√
√
2u(k) (∂i (g ij g∂j u) + ( g)bu + ( g)f )ξ 2 dx

=
B

√
(u(k) ξ)2 (∂t ( g))dx +

+

√
(u(k) )2 2ξ∂t ξ( g)dx
...

√
2u(k) ∂i (g ij g∂j u)ξ 2 dx

(A
...

B

B

Here c1 and C3 are two constants depending only on λ
...
2
...
5)
B

≤ C4 (M 2 + 1)

ξdx
...
3), (A
...
5) together yields
d
dt

√
(u(k) ξ)2 ( g)dx
B
2

≤

C 0 (B\{0}×[0,T ])
...

{u≥k}

Here C5 depends only on λ
...
Consider Kρ as in Definition A
...
Now, we integrate
the above inequality from t0 to t0 + τ to get
2

(u(k) (x, t0 + τ )ξ(x, t0 + τ ))2 dx + c1

∂i u(k) ξ 2 dx

Kρ

Q(ρ,τ )
2

(u(k) (x, t0 )ξ(x, t0 ))2 dx + C5

≤
Kρ

(u(k) )2 (|∂i ξ| + ξ |∂t ξ|)dx
Q(ρ,τ )

t0 +τ

+C4 (M 2 + 1)

ξdxdt
...
Now we pick ρ0 satisfying
C4 (M 2 + 1)(T |Kρ0 |)1/4 = 1,
so that
2

(u(k) (x, t0 + τ )ξ(x, t0 + τ ))2 dx + c1

∂i u(k) ξ 2 dx

Kρ

Q(ρ,τ )
2

(u(k) (x, t0 )ξ(x, t0 ))2 dx + C5

≤
Kρ

(u(k) )2 (|∂i ξ| + ξ |∂t ξ|)dx
Q(ρ,τ )

3/4

t0 +τ

+

ξdxdt
t0


...
5) in Chapter II of [10] for our choice of r, q, κ
...
2
there, (A
...
2) follows from it directly
...
3 (Theorem 7
...
3
...
Proof of Lemma 3
...
On the other hand,
the proof given below for B1 works for Bi (i > 1) as well with no modification at
all
...
16)
and (3
...
e
...

Note that when we scale up from B1 = B√t (p) to B, the norm of b and ∂t a becomes
smaller so the assumptions remain true
...
1
...

Assume
max |∂t a| + a−1 < C1

B×[0,1]

48

HAO YIN

for some C1
...
1)

max

B1/2 ×[3/4,1]

q

˜
|u| dV dt

|u| ≤ C2
0

B

and
1/q

1

˜ u(1)

(B
...


B

We prove (B
...
The proof is well known except that we need to
justify some integration by parts and switching of the order of integrals
...
Let qi = (3/2)i−1 q, ti = (2σ − σ 2 )(1 − 2i−1 ) and
σ
ri = (1 − σ) + 2i−1
...
Suppose that ϕi (x, t)
is a smooth cut-off function defined in Qi satisfying (1) ϕi (x, t) ≡ 0 for t = ti or
2
|x| = ri ; (2) ϕi (t) ≡ 1 for (x, t) ∈ Qi+1 ; (3) 0 ≤ ∂t ϕi + ϕ−1 ˜ ϕi ≤ σ −2 4i+2
...

+
+

˜
∂t uuqi −1 ϕi a−1 dV ds =
+
Bri ×[ti ,t]

Bri ×[ti ,t]

First, by the definition of V 2,α,[0,1] , the integral on the left hand side (hence on the
right hand side) is absolutely integrable
...
3)
Bri

t

˜
∂t uuqi −1 ϕi a−1 dsdV =
+

ti

˜
ϕi uqi −1 ˜ u + ϕi a−1 buqi dV ds
...
2 (see the footnote in the proof of Lemma 2
...

+
i

2
+qi
ti

B ri

Using the bound of ∂t a and Lemma 2
...
3) is estimated by
t
Bri

ti

t

˜
∂t uuqi −1 ϕi a−1 dsdV
+

˜
uqi ϕi a−1 dV
+

≥
Bri

−Cσ

ti

˜
uqi dV ds
...

+
Qi

2
Here we have used the definition of ϕi and estimated qi from above by 4i q 2
...


Bri+1

Bri+1

A simply way to see that the Sobolev inequality holds is to notice that the cone
metric on Bri is quasi-isometric to the flat metric on a ball of radius ri in R2
...

Integrating from ti+1 to 1, we get
qi+1 ˜
u+ dV

1

˜ uqi /2
+

≤

Qi+1

ti+1

1/2

2

+

˜
uqi dV
+

·

˜
uqi dV
+

max
t∈[ti+1 ,1]

Bri+1

Bi+1

3/2

≤

˜
uqi dV ds
+

C(q)(256)i σ −3


...


The same applies to the negative part so that (for q ≥ 2)
(B
...


There is also a standard iteration process to prove the case for q ∈ (0, 2) as we
learned from Li and Schoen [11]
...

We recycle ri and Qi by setting for i = 1, 2, · · ·
1
2
ri = 1 − i
Qi = Bri × [1 − ri , 1]
...

2i+1 − 1

We apply (B
...
1) is proved by checking the series in the above equation converges
...


50

HAO YIN

With (B
...
2 to see that there is some α > 0 and C > 0
such that
u(1) C α (B1/4 ) ≤ C u C 0 (B1/2 ×[3/4,1])
...


On the other hand, parabolic interior estimate shows that there is another C > 0
depending on V 0,α norm of a and b and C 0 norm of a−1 such that for ρ ≤ 1/4,
˜ 2 u(ρ, θ, 1) ≤ Cρ−2 u

C 0 (B1/2 ×[3/4,1])


...
2)
...
Proof of Lemma 4
...
By definition of g , there is a smooth metric g on S such that
˜
¯
g = wg
˜

on S

and
w = ρ2β
in a neighborhood of p
...
1)
which lies in Lp (S, g ) for some p > 1 because f is bounded and β > −1
...
1) because
¯
¯
wf dV =
S

˜
f dV = 0
...
By the Sobolev
¯
¯
embedding theorem, we know u is bounded and
2

¯ u 2 dV = 0
...
If u1 and u2 are two such solutions, then
u1 −u2 is a bounded harmonic function on S with respect to both the conical metric
g and the smooth metric g , which has to be a constant
...
An interpolation of Holder norm
We need several lemmas on the interpolation of H¨lder norms, which should be
o
well known
...

Lemma D
...
There is a universal constant C such that if u is in C 2,α (B) and
satisfies
u C 0 (B) ≤ 1
and
2
u(x) − 2 u(y)
[u]2,α,B := sup
≤ 1,
α
|x − y|
x,y∈B

ANALYSIS ASPECTS OF RICCI FLOW ON CONICAL SURFACES

51

then
2

u

≤ C
...
If the lemma is not true, then there exists a sequence ui satisfying
(1) ui C 0 (B) ≤ 1;
2
ui C 0 (B) ≥ i;
(2)
(3) [ui ]2,α,B ≤ 1
...
By taking subsequence if
necessary, we may assume that vi converges strongly to v in C 2 (B) norm
...


Lemma D
...
Suppose that u is a C 2,α function from B to R satisfying u
1
...
We may assume that λ := u
nothing to prove
...


is smaller than 1/8, otherwise there is
1

v(x) = λ−1 u(λ 2+α x),
1
which is defined on B := B − 2+α
...
The advantage of v is that

(D
...
2)

sup
x,y∈B

v(x) − 2 v(y)
= sup
α
|x − y|
x,˜∈B
˜y

2

u(˜) − 2 u(˜)
x
y
≤ 1
...
1 to v (restricted to B1 (x) for each x ∈ B 1

− 1
2+α
2λ

2

v

C 0 (B

1
1 λ− 2+α
2

)

) to show

≤ C,

which is exactly what we need when translated back into the inequality of u
...
Bahuaud and B
...
Yamabe flow on manifolds with edges
...

[2] X
...
Wang
...

a
arXiv:1402
...

[3] X
...
Wang
...

a
Journal of Functional Analysis, 269(2):551–632, 2015
...
Donaldson
...
In Essays in mathematics
a
and its applications, pages 49–79
...

[5] D
...
Trudinger
...
SpringerVerlag Berlin, 1983
...
Hamilton
...
Contemp
...

[7] T
...
Uniqueness of K¨hler-Einstein cone metrics
...

[8] T
...
Loya
...
International
Mathematics Research Notices, 2003(3):161–178, 2003
...
Jeffres, R
...
Rubinstein
...

a
Annals of Mathematics, 183:95–176, 2016
...
A
...
A
...
N
...
Linear and quasilinear equations of parabolic type, Translations of Mathematical Monographs, Vol
...
American Mathematical Society, Providence, 1968
...
Li and R
...
Lp and mean value properties of subharmonic functions on Riemannian
manifolds
...

[12] G
...
Second order parabolic differential equations
...

[13] R
...
Elliptic theory of differential edge operators I
...

[14] R
...
Rubinstein, and N
...
Ricci flow on surfaces with conic singularities
...

[15] E
...
Heat kernel asymptotics on manifolds with conic singularities
...

[16] D
...
Song, J
...
Wang
...

arXiv:1407
...

[17] D
...
Song, J
...
Wang
...
arXiv:1503
...

[18] D
...
Smoothening cone points with ricci flow
...
5554, 2011
...
Simon
...
In Proceedings of the International Conference held to honour the 60th Birthday of
AM Naveira, volume 8, page 14
...

[20] M
...
Local smoothing results for the Ricci flow in dimensions two and three
...

[21] P
...
Ricci flow compactness via pseudolocality, and flows with incomplete initial
metrics
...

[22] M
...
Prescribing curvature on compact surfaces with conical singularities
...

[23] Y
...
Smooth approximations of the Conical K¨hler-Ricci flows
...
5040, 2014
...
Yin
...
Journal of Geometric Analysis,
20(4):970–995, 2010
...
Yin
...
arXiv:1305
...

[26] H
...
Zheng
...
preprint, 2016
...
edu

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Title: CRF
Description: a note which instruct the research result on calabi flow,including the short time existence,the stable property.

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