- UPPER BOUND FOR THE LOCAL ENTROPY fsvdμ 199
Since t::::; c^2 , this implies that for (x, t) EM x [o, c^2 ] where¢/ (s (x, t))-=/=
0,(8 ) (n-1)(2a+7)
at - b.g(t) dg(t) (x, xo) 2: - 3Vt
(22.53)
lOOn
>---- Vt 'which holds in the weak sense; for the last inequality we used a ::::; 1. Using
a = 200n and by substituting (22.53) into (22.48), we have that for all
(x, t) E M x [o, c^2 ] (independent of whether or not <// (s (x, t)) is equal to
zero)
¢"(s(x,~) 2
Oh (x, t) ::::; - b 2 l\7dg(t) (x, xo) lg(t)
10
::::; b 2 h(x, t)in the weak sense, since¢' (s):::; 0, -<f/'(s):::; 10¢(s), and
l\7d 9 (t) (x,xo)l~(t) = 1
a.e. This completes the proof of the lemma. D
3.2. A negative upper bound for the local entropy at time zero.Let (Mn, g (t)), t E [O, c^2 ], be a complete solution of the Ricci fl.ow with
bounded curvature and let (x, f) EM x (0, c^2 ]. Define the function
(22.54) H (x, t) ~ (47r (f - t))-n/^2 e-f(x,t)on M x [O, t) to be the minimal positive fundamental solution of the adjoint
heat equation centered at (x, f), i.e.,
(22.55a) D* H ~ (-:t -b. 9 + R 9 ) H = 0,
(22.55b)We have
JM H (x, t) dμg(t) = 1.
Define Perelman's Harnack quantity v on M x [O, t) by
(22.56) v (x, t) ~ ( (t - t) ( R 9 + 2b. 9 f - J\7 9 fJ^2 ) + f - n) H.
Here is the main result of this section.
LEMMA 22.13. Let (Mn,g(t),xo), t E [O,c^2 J, where c E (O,co], be a
complete pointed solution to the Ricci flow with bounded curvature such that
a 2
(22.57) I Rm I (x, t) ::::; - + 2 whenever t E (0, c^2 ] and dg(t) (x, xo) ::::; co,
t co