1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

3. EXTENSIONS OF HAMILTON'S COMPACTNESS THEOREM 141

3.3. Compactness for solutions on orbifolds. Note that Defini-

tions 3.5 and 3.6 can be easily generalized to orbifolds. (See [343] for the

definition of orbifold.) We have the following generalizations of Theorems

3.9 and 3.10 to orbifolds. The version for metrics is

THEOREM 3.24 (Compactness theorem for metrics on orbifolds). Let

{(Mk', gk, Ok)} be a sequence of complete pointed Riemannian orbifolds and

let L:k be the singular set of Mk· Suppose that

(i) [V}; Rmk [k :::; Gp on Mk for all p 2: 0 and k, where Gp < oo are

constants independent of k, and

(ii) Volgk Bgk (Ok, ro) 2: vo for all k, where ro > 0 and vo > 0 are two

constants independent of k.

Then either of the following hold.

(1) limk---+oodgk(Ok,L:k) > 0. In this case there exists a subsequence

{ ( Mki, gkj, Oki)} which converges to a complete pointed Riemannian orb-
ifold (M~,goo,0 00 ) with [\7~ 00 Rmg 00 [g 00 :S Gp andVolg 00 Bg 00 (0oo,ro) 2:
vo. Furthermore 000 is a smooth point in M 00 •
(2) limk---+oodgk (Ok, L:k) = 0. In this case there exists a subsequence
{(Mkj' gkj' Okj)} such that limj---+oo d(Okj, L:kj) = 0. If we choose O~j E L:kj

with dgk ( okj' O~) = d( okj' L:kj)' then a subsequence of { (Mkj' gkj' o~J}

converges to a complete pointed Riemannian orbifold (M 00 , g 00 , 000 ) with

[\7~ 00 Rmg 00 [g 00 :S Gp and Volg 00 Bg 00 (0 00 , ro) 2: vo. Furthermore 000 is a

singular point in M 00 •

The version for solutions of Ricci flow is the following.

THEOREM 3.25 (Compactness theorem for solutions on orbifolds). Let

{(Mk', gk(t), Ok)}, t E (a, w), be a sequence of complete pointed orbifold

solutions of the Ricci flow. Let L:k be the singular set of Mk· Suppose that

(i) [Rm k [k :S Co on Mk x (a, w) for all k, where Co < oo is a constant

independent of k, and

(ii) Volgk(o) Bgk(o)(Ok, ro) 2: vo for all k, where ro > 0 and vo > 0 are

two constants independent of k.

Then either of the following hold.

(1) limk---+oodgk(o)(Ok,L:k) > 0. In this case there exists a subsequence

{ (Mkj, gkj (t), Okj)} which converges to a complete pointed orbifold solution

of the Ricci flow (M~,goo(t),Ooo) with [Rmg 00 [g 00 :S Co on Moo X (a,w)

and Volg 00 (o) Bg 00 (o)(0 00 ,ro) 2: vo. Furthermore 000 is a smooth point in

Moo.

(2) limk---+oodgk(o)(Ok, L:k) = 0. In this case there exists a subsequence

{(Mkj,gkj(t), Oki)} such that limj---+oo d(Oki' L:kj) = 0. Furthermore if we

choose O~i E L:ki with dgk (0) (Oki, O~i) = dgk (0) (Oki, L:ki), then there is a

subsequence of { (Mkj, gki ( t), o~J} which converges to a complete pointed

orbifold solution of the Ricci flow (M~, goo(t), Ooo) with [Rm g 00 [g 00 :S Co