- LI-YAU ESTIMATE FOR POSITIVE SOLUTIONS OF HEAT EQUATIONS 329
and
(2) (uniform bounds on its first and second covariant derivatives with
respect to g (0))(25.91) IV' flg(O) (x) :S Cn,K and IV'g(O)V' Jlg(O) (x) :S Cn,Kfor all x E M, where Cn,K E (1, oo) depends only on n and K.
Since f 7 9ij = 2Rij and since (25.51) implies
we have
(25.92)and
-Agij (x, T) :::; Rij (x, T) :::; Agij (x, T) on M x [O, T],e-^2 AT g (x, 0) :::; g (x, T) :::; e^2 AT g (x, 0)
(25.93) e-AT dg(O) (x,p):::; dg(T) (x,p):::; eAT dg(O) (x,p)for all (x, T) EM x [O, T].
Hence (25.90) implies
e-AT dg(T) (x,p) + 1:::; f (x) :::; eAT d 9 ( 7 ) (x,p) + Cn,K·
The bounds (25.91) and (25.92) also imply that on M x [O, T]
(25.94) IV' f lg(T) (x) :S eAT Cn,K
and
(25.95)
where C4 < oo depends only on n, K, A, T, and supMx[O,T] IV'iRjkl· Here
we used (25.91),
V'f(T)V'jf - V'f(O)V'jf = (rfj (0) -rfj (T)) \i'kf,
and
\rfj (o) - rfj (f)\ = 11
7
d~ rfj (T) dTI:'S 31
7
\\i'f(T)R,jk\ (T) dT.Note that since f is independent of time, (25.95) implies
I-~~+ Llfl:::; y'nC4.
By (25.89) now we havea¢ ( n ) IV'</>12
8T - Ll</> +^2 + 2 (2 - 3c) c -¢-(25.96) :::; ~vnC4 VG¢ c + ( 3 + n ) {) 2AT^2.
2 ( 2 _ 3c)c R^2 e Cn,K ::::;= C3