1547671870-The_Ricci_Flow__Chow

2. POSITIVE CURVATURE IS PRESERVED 255

( c) a cigar product (IR^3 , g ( t)), where g ( t) is the self-similar solu-
tion corresponding to the soliton metric introduced in Subsec-
tion 2.1.
For any limit in Case 2a, one could perform a technique called dimen-
sion reduction in the hope of obtaining an ancient solution. (Dimension
reduction is introduced in Section 4 below and will be discussed further in a
planned successor to this volume. It involves taking a limit around a suitable
sequence of points tending to spatial infinity.) If the ancient solution one
obtains is not in fact an eternal solution which attains its maximum curva-
ture, one can then take a third limit about a suitable sequence of points and

times tending to -oo where the curvature is sufficiently near its maximum.

Having done so, one expects to see either Case 2b or 2c above.
If one were in Case la, then one could conclude from compactness of the
limit that the underlying manifold of the original solution must have been
S^3 or one of its quotients. In the other cases, the singularity model should
give local information about the original solution near the singularity just
prior to its formation.

REMARK 9.1. The recent work [105] of Perelman rules out Case 2c for
singularities that form in finite time.

2. Positive curvature is preserved

In this chapter, we shall make heavy use of the maximum principles for

systems discussed in Section 3 of Chapter 4. In so doing, we will follow

the methods introduced in Section 4 of Chapter 6. In order to review these

techniques, we will first apply them to derive two results which were obtained

earlier by classical methods in Sections 1 and 3 of Chapter 6.

LEMMA 9.2 (Positive sectional curvature is preserved.). If (M^3 ,g(t))

is a solution of the Ricci flow such that the initial metric g(O) has posi-

tive (nonnegative) sectional curvature, then the metrics g(t) have positive

(nonnegative) sectional curvature for all t > 0 that the solution exists.

PROOF. We show that positive sectional curvature is preserved at each

x E M^3 ; the proof that nonnegative sectional curvature is preserved is

entirely analogous. Let >. (lfD) 2 μ (lfD) 2 v (lfD) denote the eigenvalues of any

lfDE (A^2 TM^30 s!\^2 TM^3 )x.

Let M (t) E (A^2 T* M^3 @s !\^2 T* M^3 )x be the quadratic form corresponding

to Rm [g]. Define the subset Kc (A^2 T* M^3 @s !\^2 T* M^3 )x by

K ~ {lfD : v (lfD) > O},

and notice that M (t) E K if and only if all sectional curvatures are positive

at (x, t). It is easy to see that K is invariant under parallel translation. K

is convex in each fiber because if

P: (A2T*M3 0s !\2T*M3)x .___.IR