1547671870-The_Ricci_Flow__Chow

NOTES AND COMMENTARY 221

for all positive time.

From here it is not hard to show that all derivatives of the curvature
decay exponentially. This implies that g ( t) converges exponentially fast in
every cm norm to a smooth Einstein metric g=; for details, the reader is
referred to Section 17 of [58]. (Also see Section 5 of Chapter 5.)

Notes and commentary

The pinching estimate in Lemma 6.28 has an analogue for the mean

curvature flow [74]. This is also true for Theorem 6.30; however, the proof

of the analogous estimate for the mean curvature flow requires integral es-

timates. In dimension n = 4, pinching estimates for the Ricci flow we

obtained by Hamilton under the assumption of positive curvature opera-

tor [59]. Higher dimensional estimates under stronger pinching assump-

tions were obtained independently by Huisken [75], Margerin [95, 96], and

Nishikawa [102, 103].