1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

40 18. GEOMETRIC TOOLS AND POINT PICKING METHODS

In §4 we discuss rough monotonicity of the size of necks in complete
noncompact manifolds with positive sectional curvature. This will be useful
in the study of 11;-solutions.
Finally, in §5 we discuss a local form of the weakened no local collapsing
theorem presented in Chapter 8 of Part I.

1. Estimates for changing distances

Now we turn to some geometric and intuitive aspects that we borrow

from the study of manifolds with nonnegative sectional curvature and which

influence the study of Ricci flow, including 11;-solutions. By definition, 11;-

solutions have nonnegative curvature operator, which implies nonnegative

sectional curvature. Since compact 3-dimensional 11;-solutions must have

positive sectional curvature and hence are diffeomorphic to spherical space

forms, we are most interested in noncompact 11;-solutions.

In relation to nonnegative sectional curvature, Greene wrote the follow-

ing on p. 100 of [7 4]:

'Nonnegativity, and especially positivity, of curvature tends

to make long geodesics nonminimizing. Thus some tension

arises. between the necessary existence of rays and the curva-

ture's nonnegativity.'

Greene went on to write:

'The fact that straight lines in ~n are, in this sense, just

barely minimizing suggests that there should be quantitative

estimates on just how much positivity of curvature could be

possible without forcing a geodesic to be nonminimal.'

An important illustration of this idea is how much positivity of Ricci

curvature a minimal geodesic ry can have. Namely, in Proposition 18.8 below

we shall obtain an upper bound for f 7 Rc (ry' (s) ,ry' (s)) ds. An application

to Ricci flow of this bound is given in Lemma 8.3(b) of Perelman [152],

which discusses the change in the distance function under the Ricci flow

(see Theorem 18.7(2) below). The precursors for this result are Theorems

17.2 and 17.4 of Hamilton [92]. See also Proposition 1.94 as well as Lemma

8.33 and Remark 8.34, all in [45].

1.1. The time derivative of the distance function.

First we recall the formula for the time derivative of the distance function

using the lim inf of forward difference quotients. Our presentation follows

the proof of Lemma 3.5 in Hamilton [89] (see also Lemma 10.29 in Part II).

For a function f (x, t), let

{)-;:;i f ( x, t) -;--'--l" 1n11n. f f ( x' t + h) h - f ( x' t)

ut h-tO+