1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

88 19. GEOMETRIC PROPERTIES OF K:-SOLUTIONS

There are two classes of solutions to Ricci flow which have infinite as-

ymptotic scalar curvature ratio.

THEOREM 19.23.

(1) (Steady gradient Ricci soliton) If (Mn,g (t), \J f (t)), t E (-oo, oo),

n ~ 3, is a complete steady gradient Ricci soliton with sect (g ( t)) >

0 and with the property that R attains its maximum at some point,

then ASCR (g (t)) = oo.

(2) (Type I ancient solution) If (Mn,g(t)) is a complete noncompact

Type I ancient solution with bounded positive curvature operator,

then ASCR(g(t)) = oo.

For (1) see the original Theorem 20.2 in [92] or Theorem 9.44 in [45]

and for (2) see Theorem 9.32 in [45].

Regarding backward limits of Type II ancient solutions,^7 we have the

following (see Proposition 9.29 in [45]).

THEOREM 19.24. If (Mn, g(t)) is a complete noncompact Type II ancient

solution with Rm ~ 0 and bounded positive sectional curvature, then there

is a backwards limit which is a steady gradient Ricci soliton which attains

its maximum scalar curvature.

This result and Theorem 19.23(1) may suggest the following.

PROBLEM 19.25. If (Mn, g(t)) is a complete noncompact Type II ancient

solution with bounded positive curvature operator, then is ASCR(g(t)) =

oo? Here we are not assuming that the solution is K,-noncollapsed at all

scales.

PROBLEM 19.26. Let (Mn,g(t)) be a complete noncompact ancient so-

lution and let (M~, g 00 (t)) be a backwards limit. Is ASCR (g (t)) = oo

if and only if ASCR (g 00 (t)) = oo? More ambitiously, one may ask if

ASCR (g 00 (t)) = ASCR (g (t)).

In this chapter and the next chapter we shall use the invariants AVR

and ASCR to study K,-solutions.

2.2. K,-sollutions with Harnack.

We now discuss a variant on the notion of K,-solution, which we shall find

useful in formulating Perelman's compactness theorem in higher dimensions;

see § 4 of this chapter and §3 of the next chapter.^8

DEFINITION 19.27 ( K,-solution with Harnack). If the bounded curvature

condition (ii) in Definition 19.7 is replaced by the requirement that g (t)

(^7) By definition, Type II (backward in time) means that supMx(-oo,o] !ti !Rml (x, t) =
00.
(^8) Definition 19.27 was introduced by one of the authors [142]; see also p. 406 in Part
I.