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Note of Ricci flow on manifold with conical
singularities
Hu xiyu
October 15, 2016

1

some eg and observations

Eg 1 (M 2 , g),g(t) = e2u(t) g0 ,
r
∂u
˜
= e−2u ∆u + − e−2u K0
∂t
2
Eg 2 (M n , gij (t))
∂gij (t)
= −2Ric(gij )
∂t
Observation 1 (”magic theorem”) The given ”smooth” initial :
∃ T small ,T > 0,the solution exists on [0, T ]
Observation 2 equation is possible system
...

Observation 5 ”geometry”

1

2

smooth manifold with conical singularities

on surface we can define conical singularity
...

iff conical background metric g0 : g0 near p,g0 = r2β (dr2 + r2 dθ2 ),(r, θ) is
the interpolation coordinate chart
...


3

rough line of proof

initial u0 ,

∂u
r
˜
= e−2u ∆u + − e−2u K0
∂t
2

Step 1:(Short time existence)
state and proof the ”magic theorem”:
1
...

2
...
maybe use shauder fix
point theorem or contraction map theorem or else
...
so the problem reduce to get this type estimate,
r
˜
if ∂ui+1 = e−2ui ∆ui+1 + 2 − e−2ui K0
...
T[0,T ] is continuous,Dom(T[0,T ] )
is convex,Im(T[0,T ] ) is a pre-compact set
...

Step 2:(Threshold type theorem,long time existence)
Threshold as long as ||u||L∞ is bounded
...

Basically is based on Alzalo-Ascoli theorem
...
the question does not existence for smooth manifold
...
singular space
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what is the optimal regularity?
the problem naturally come from both ”pure PDE” and ”application for
geometry problem”
...
Wang]
Donaldson setting C 2,α,β

4

More seriously treat with the problem

in 07 years,consider the problem
Model 1
∂u
= ∆u
∂t
on M − {p}
u|t=0 = f
where f is a function with nice regularity
...
and then use the theory of operator semigroup
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remark:the extension is not unique so the information we know for the solution is very little
...
so the treat of Functional analysis is
not enough for us
...

we set Mi is the manifold cut off form M with a boundary more and more
near the singularity
...
e
...

Under the general setting this become:
Model 3
∂uk
= a(k, t)∆uk + b(k, t)∂ i uk + c(x, t)
∂t

∂uk
|∂Mk = 0
∂v
when k −→ ∞, do we have uk −→ u?
we need priori estimate: Schauder estimate for serious parabolic equation
tell us:
for equation ∂u = ∆u on M with priori estimate ||u||L∞ ≤ C, we have:
∂t
|

k

u(p)| ≤

C
rk

wher r is the maximal such that geodesic ball B(r, p) ⊂⊂ Mi
...

1
...
(now we do not know what the norm || · ||∗∗∗
need to be)
we have C 0 estimate and the energy estimate as follows:
from maximal principle,easy to get C 0 norm estimate
...


from energy method we can estimate

Mk

|| uk ||2
...

so we get:
∂
|uk |2 ≤
| f |2 +
∂t Mk
Mk

Mk

| uk |2

so we can bounded Mk |uk |2
...

∂v
otherwise we will get solution u ∈ W 1,2 (Mk ) ∩ C 2 (Mk )
...

in this case to prove the short time existence we need follow four claims is
ture
...

first cover the whole manifold by a open set have positive distance t=with the
d
conical singularity and a countable group of set An = B( 2d , p)−B( 2n+1 , p),which
n
is balls center at singularity p with radius 2d
...
e
...
,∞ ||f (2−k , θ)||C k,α (B1 −B 1 ) + ||f ||C k,α (U )
2

easy to see the definition is independent with the cover and the local
interpolation coordinate chart
...

that is
δu = f
on S − {p}
...
then
||u||εk+2,α ≤ C(||u||L∞ + ||f ||εk,α ) ≤ C(C1 + ||f ||εk,α )
6

in fact we only need to add each inequality come from each open set of the
cover by Schauder estimate to proof this
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Definition 3 (| · |w )
1

˜
| ˜ u|2 dV ) 2

|u|w = (
S

Definition 4 (W k,α ) the set of all f in εk,α with finite |f |w ,
||f ||k,α = ||f ||εk,α + |f |w
W
in Banach space
...
,∞ ||f (2−k ρ, θ, 4−k t)||C l,α ((B1 −B 1 )×[0,4−k T ]) +||f ||C l,α (U ×[0,T ])
2

from the definition,easy to see
∂u
= ∆u + f
∂t
om M
u|t=0 = u0
=⇒
||u||ρl+2,α,[0,T ] ≤ C(||u0 ||εl,α + ||f ||ρl,α,[0,T ] + ||u||C 0 (S×[0,t]) )
easy from the classical schauder estimate
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weak sense:
1
...
|u|v < +∞

8



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