1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

2. THE 1"-NONCOLLAPSED CONDITION

satisfies the trace Harnack estimate

8R ·

(19.11) 8t + 2 (\/ R, X) + 2 Re (X, X) 2: 0

89

for all vector fields X, then we say that g (t) is a K-solution with Harnack.

By taking X = 0 in (19.11), we immediately see that, for a K-solution

with Harnack, the scalar curvature R is pointwise nondecreasing in time,

i.e., 9Jf-2: 0. Thus, if a K-solution with Harnack g (t) defined fort::::; 0 does

not have bounded curvature in space-time, then

sup R (x, 0) = oo.

xEM

In this case we clearly have ASCR (g (0)) = oo.

Note that by Hamilton's trace Harnack estimate, i.e., Corollary 15.4

in Part II, a K-solution is a K-solution with Harnack. The only possible

distinction between these two notions is when M is noncompact and the

curvatures of g (t) are unbounded.

REMARK 19.28. A reason for why the notion of K-solution with Harnack

is useful is that under certain limits, the condition that the trace Harnack

estimate holds is preserved while, a priori, the condition that the curvature

is bounded may not be preserved.

Perhaps one may improve Hamilton's matrix Harnack estimate in the
following direction.
PROBLEM 19.29 (Hamilton's matrix Harnack estimate without a bound
on Rm). In Hamilton's matrix Harnack estimate, i.e., Theorem 15.1 in Part
II, can one weaken the part of the hypothesis requiring the curvature to be

bounded? In particular, is there a class of complete solutions (Mn, g (t)),

t E [O, w), to the Ricci flow with Rm (9 (t)) 2: 0 and supM !Rm (9 (t))I = oo

for each t E (0, w) for which Hamilton's matrix Harnack estimate holds for
solutions in this class? Perhaps one may assume that the curvatures satisfy
a growth condition in terms of the distance function to a fixed point.

A simpler question than Problem 19.29 may be whether Hamilton's trace
Harnack estimate holds for a class of solutions on surfaces with unbounded
curvature. As far as we know, not much is known about complete solutions
to the Ricci flow on noncompact surfaces with unbounded curvature. A
related problem is that of 'localizing' the Harnack estimate. In the case of
the heat equation on a fixed Riemannian manifold, this was accomplished
by Li and Yau [121].

REMARK 19.30 (Harnack estimate when Rm is bounded but not uni-
formly bounded). Note that if (Mn, g (t)), t E [O, w), is a complete solution

to the Ricci flow such that Rm (g (t)) 2: 0 and supMx(to,w) !Rm (g (t))I < oo

for each to E (0,w),^9 then Hamilton's matrix Harnack estimate holds for

(^9) Note that here we have not assumed a uniform (in time) bound on the curvatures.