76 18. GEOMETRIC TOOLS AND POINT PICKING METHODS
IRc(z, t) I ::;; n-1 for all z E Bg(t) (xo, ~e^1 -n), we can apply Theorem 18.7(1)
to obtain
(18. 73)
whenever (w, t) has ¢/(u) i=-0.
Recall from Lemma 7.46 in Part I that L 7 + b.L ::;; 2n. Applying (18.66)
and (18.73) to (18.72) presuming¢/ (u) i=- 0 (if¢' (u) = 0, then this follows
from (18.72) directly), we find that at any point (y,t) where both h(y, t) =
H ( t) and t ?:. ~
(gt -b.) h(y, t) ?:. -2n¢(u) - (L + 2n + 1) C1 (n, A)¢(u)
?.-(2n+C1(n,A))h(y,t).
Here we have used L(y, r) ?:. -2n for T ::;; ~ to obtain the last inequality.
This proves part ( 4) of the claim and hence both Lemma 18.38 and Theorem
18.36. D
REMARK 18.39. Regarding the standard issue of the nonsmoothness of
h, see the remark at the end of §2 in Chapter 25.
6. Notes and commentary
§2. There are several other point picking lemmas used in the Ricci flow
besides the ones we give in this chapter. For example, see
(1) the point picking results in §1 of Chapter 22, which are used in the
proof of the pseudolocality theorem,
(2) the point picking result in the proof of Theorem 12.1 in Perelman
[15~:].
Now we raise a general question.PROBLEM 18.40 (General point picking). Formulate Bourbaki-type re-
sults about point picking (in particular, show general results exhibiting the
essences of point picking). It would be desirable for the results to include
the results in this section as well as other point picking results in Ricci flow,
in particular, the methods in Perelman [152] and Hamilton [92]. Perhaps
such a result may include phrases such as the following for example. Let S
be a metric space, let f : S -+ JR be a function, and let ¢ : S -+ JR be a
reference function, etc., where examples off are Rand IRml and examples
of¢ are d (·, 0)^2 and T-t.
In contrast to the nontrivial limits we get from the above methods of
point picking, consider the following pathological case.
EXERCISE 18.41 (Trivial limits). Show that for any sequence of pointed