1547671870-The_Ricci_Flow__Chow

l. GLOBAL ESTIMATES 225

Before proving Theorem 7.1, we will calculate the evolution of the square
of the norm of the curvature tensor. Besides being a prototype for the sort
of equations we will encounter in the proof of the theorem, this result has
useful applications of its own.

LEMMA 7.4. If (Mn, g (t)) is a solution of the Ricci flow, then the square

of the norm of its curvature tensor evolves by

gt 1Rml^2 =~1Rml^2 - 2 l\7Rml^2

• 4grigsjgpkgqf!Rrspq (Bijkl! - Bijl!k + Bikjl! - Bil!jk)'

where

In particular, one has

gt 1Rml^2 :S ~ 1Rml^2 - 2 l\7 Rml^2 + C 1Rml^3 ,

where C is a constant depending only on the dimension n.

PROOF. By formula (6. 17 ), the ( 4, 0)-Riemann curvature tensor evolves

by

(7.la)

(7.lb)

a

at Rijke = ~Rijke + 2 (Bijke - Bijl!k + Bikj€ - Biejk)

• (Rf RpjkR + R~Ripki! + R~RijpR + R~Rijkp).

It is easy to check that

at a IR m^12 - _ at .!!_ ( g g ri sj g pk g qf R rspq R i1kR. )

= 2grigsjgpkgqeRrspq [~RijkR + 2 (BijkC - BijRk + BikjR - BiRjk)];

indeed, the terms that arise when we differentiate g -^1 exactly cancel the

terms that appear in line (7.lb). Since

~ 1Rml^2 = 2grigsj gPkgqC Rrspq~RijkC + 2 l\7 Rml^2 ,

the result follows. D

The following important consequence of Lemma 7.4 explains why the

assumption in Theorem 7.1 that IRml is bounded for a short time is a

reasonable one.

COROLLARY 7.5 (Doubling-time estimate). There exists c > 0 depending

only on the dimension n such that if (Mn,g (t): 0 :St :ST) is a solution of

the Ricci flow on a compact manifold and

M (t) =i= sup IRm (x, t)lg(x,t),

xEMn