1547671870-The_Ricci_Flow__Chow

274 9. TYPE I SINGULARITIES

If (9.12) holds, then by Lemma 9.12 there is rt (6) such that
0
P 2: rt 1Rml^2 1Rml^2 ,
whence it follows that taking c: :S: rt gives
J < - rye: G

• 4lt1.
  Thus in either case, G is a subsolution of the heat equation for all times


• oo < t < T , because

~G < t:::.G^2 (l - c:) (VG '\! R) - rye: G.

at - + R ' 2 iti

0

6. Type I ancient solutions on surfaces
   The objective of this section is to prove that the shrinking round 2-
   sphere is the only nonflat Type I ancient solution of the Ricci flow on a
   surface. Because of dimension reduction, this fact plays an important role
   in the classification of 3-dimensional singularities.
   Before we classify Type I ancient solutions on surfaces, we shall establish
   a result which is of considerable independent interest. It follows from a gen-
   eralization of the LYH differential Harnack estimates introduced in Section
   10 of Chapter 5. We will discuss the full family of LYH differential Harnack
   estimates for the Ricci flow in a chapter of the successor to this volume. For
   now, we simply state the following result from [61].

PROPOSITION 9.20 (LYH estimate, trace version). If (Mn,g (t)) is a

solution of the unnormalized Ricci flow on a compact manifold with initially

positive curvature operator, then for any vector field X on Mn and all times

t > 0 such that the solution exists, one has

a R

0 :S: at R + t + 2 ('\! R, X) + 2 Re (X, X).

COROLLARY 9.21. If (Mn, g (t)) is a solution of the unnormalized Ricci

flow on a compact manifold with initially positive curvature operator, then

the function tR is pointwise nondecreasing for all t 2: 0 that the solution

exists. If (Mn, g ( t)) is also ancient, then R itself is pointwise nondecreasing.

PROOF. Taking X = 0, we have

at a (tR) = t (a at R + i R) ;:::: o

for all t 2: 0 such that the solution exists. This proves the first assertion.

To prove the second assertion, notice that one has

a R

-R+-->O

at t+a -