1547671870-The_Ricci_Flow__Chow

232 7. DERIVATIVE ESTIMATES

of the Ricci flow existing for t E (a , w) with the properties that

sup [sect [goo] I :::::; K

M~x(a,w)

and

inj 900 (0) (Ooo) 2: 5.

The convergence is in the following specific sense: there exist a sequence

of open sets Ui ~ M~ such that 000 E Ui for all i E N and such that

any compact subset of M~ is contained in Ui for all i sufficiently large,

a sequence of open sets Vi E Mi such that Oi E Vi for all i E N, and

a sequence of diffeomorphisms 'Pi : Ui __, Vi such that 'Pi (Ooo) = Oi and

('Pi)* (F 00 ) =Fi for all i EN; these diffeomorphisms have the property that

the pullbacks 'Pi (gi) converge uniformly to g 00 in every cm norm on any

compact subset of M~ x (a, w).

N ates and commentary

Two good references for derivative estimates that apply to general par-

abolic equations are [94] and [89]. The reader may also be interesting in

consulting some of Bernstein's original papers, notably [16, 17, 18]. For

the Ricci fl.ow in particular, the early Bemelmans- Min-Oo-Ruh paper [11],

Bando's paper [9], and Shi's papers [117, 118] are all useful references. Our

approach is a modification of the method outlined in Section 7 of [63].

The convergence theory of Cheeger and Gromov and in particular the

Gromov- Hausdorff convergence of Riemannian manifolds is currently an ac-

tive and rich area of research. An interested reader is urged to consult the

influential book [53] and the fine survey [109]. Good examples of conver-

gence results in the literature can be found in [2], [26], [45], [50, 51], [108],

and [130, 131].