198 22. TOOLS USED IN PROOF OF PSEUDOLOCALITY
where a :S 1. If(22.46a)
(22.46b)a= 200n,
b 2: 5EoA,
where A;::: 41n, then h is a subsolution to the following linear heat equation:(22.47) Oh (x, t) ~ ( :t - b.g(t)) h (x, t) :S ~~ h(x, t) on M x [o, c
2
]in the weak sense.
PROOF. Let s(x,t) ~ dg(t)(x,~o)+av't, so that h(x,t) =
compute the heat operator acting on h as
(22.48) Oh (x, t) = 1>' (s ~x, t)) ( ( :t - b.g(t)) dg(t) (x, xo) +
2~)1>" (s (x, t)) I
12- b 2 Y'dg(t) (x, xo) g(t).
We shall now apply Perelman's lower bound for (gt - b.g(t)) dg(t)· Firstnote that at any point (x, t) E M x [O, c^2 ] where ¢/ (s (x, t)) -/=-0, we have
s (x, t) E [1, 2], which implies
(22.49) b - aVt :S dg(t) (x, xo) :S 2b - aVt :S 2b.
In particular,
(22.50) supp (h (·, t)) c Bg(t) (xo, 2b).
Since (22.46) and A ;::: 41n imply that a and b satisfy(22.51) b;::: (a+ 1) co,we have
(22.52) b - aVt;::: Vt for all t E [o, c^2 ].
Then h(x, t) = 1 for x E Bg(t) (xo, Vt) and t E [o, c^2 ].
The hypothesized curvature bound (22.45) implies
where
Re :S (n - 1) K in Bg(t) (xo, ro),a 2
ro = Vt :S E and K = - + -.
t c§By Theorem 18.7(1) (see also the original Lemma 8.3(a) in Perelman [152]),
if a point (x, t) EM x [o, c^2 ] is such that¢' (s (x, t)) -/=-O, then
(:t-b.g(t))dg(t)(x,xo)2'.-(n-l)(~(T+ :6
)Vi+ ~);
note that (22.52) implies x EM - Bg(t)(xo, ro).