188 22. TOOLS USED IN PROOF OF PSEUDOLOOALITYWe note the following elementary variant of Perelman's Claim 1.EXERCISE 22.5. Let (Mn, g (t)), t .E [o, c^2 J, where c E (0, co], be a
continuous family of complete smooth Riemannian manifolds, let xo EM,and let S c M x (0, c^2 ]. Show that if there exists (x1, t1) E S with x1 E
Bg(ti) (xo, co) and I Rm I (x1, t1) > z~, then there exists (x,f) ES with
(22.15) x E Bg(f) (xo, (2A + l)co) and f E (0, t1],
such that I Rm I ( x ,t) > co^12 and(22.16) I Rm I (x, t) :S 41 Rm I (x, f)
for all
(22.17)
(x, t) ES with 0 < t :Sf and dg(t) (x, xo) :S dg(f) (x, xo) +Al Rm 1-~ (x,t).
In preparation for Claim 2 on p. 24 of [152] we note the following.REMARK 22.6. We wish to show that all points in a parabolic cylin-
der centered at (x, f) (not centered at (xo, f)) satisfy (22.8). In view of
the changing distances estimate, we consider the more ideal case where
Re (g (t)) :S 0 for t E [o, c^2 ], so that distances are nondecreasing in time.Lett E [O,~.
(1) If dg(f) (x, x) :S Al Rm 1-~ (x, f), then by the triangle inequality,
dg(t) (x, xo) :S dg(f) (x, xo)
:S dg(f) (x, x) + dg(f) (x, xo):S dg(f) (x, xo) +Al Rm 1-~ (x, f).
1 -
(2) Alternatively, if d 9 (t) (x, x) :S Al Rm 1-2 (x, t ), then
dg(t) (x, xo) :S dg(t) (x, x) + dg(t) (x, xo):S d 9 (f) (x, xo) +Al Rm 1-~ (x, f).
Hence Claim 1 implies that (22.8) holds for all points (x, t) E Ma satisfying
the hypothesis of either (1) or (2).1.2. The existence of a well-chosen large curvature point.
Now we show that the point picking method in the previous subsection
gives a well-chosen large curvature point, i.e., the curvature estimate (22.8)
holds inside a parabolic cylinder of (x, f). The following is Claim 2 on p. 24
of [152].
Claim 2 (The well-chosen large curvature point (x, f) has con-
trolled curvature in a backward parabolic cylinder centered at
(x, f)). Under the assumptions of Claim 1, further assume A ;:::: 1 and
o: < 13 (n~l)vn· For the point (x, f) shown to exist in Claim 1, we have
(22.18) I Rm l(x, t) :S 41 Rm l(x, f)