5. THE SECOND VARIATION FORMULA FOR £ AND THE HESSIAN OF L 319

If L (·, f) is not C^2 at q, the above inequality is understood in the bar-

rier sense; this is our convention below. More precisely, there is a smooth

function L (·, f) defined near q such that ( Hess(q,-r) L) (Y, Y) satisfies in-

equality (7.70) and L (-, f) is an upper barrier function for L (·, f) with

L(q,r)=L(q,r).

LEMMA 7.41 (Upper bound for Hessian of L). Fix To E (0, T). Given

f E (0, To], q E M, and Y E TqM, the Hessian of the L-distance function

L (·, f) at q has the upper bound

(7.71) ) (

d;(-r) (p, q) 1 ) 2

(Hess(q,7') L (Y, Y) :::; C2 + C2 VT + VT JYI '

where C2 is a constant depending only on n, To, T, Co (Co 2':. sup Mx[O,T] IRml

is as in (7.27)).

PROOF. From Shi's derivative estimate and the equation for g 7 Re, there

is a constant C1 depending on n, T-To, and Co such that I g 7 Rel , l\7\7 RI ,

IV' Rel , and l\7\7 Rml are all bounded by C1 on M x [O, To]. From (7.52)

and Lemma 7.13(ii) and (iii), we get

I 'TX (T)l2 < e6G

(^0) T I !TX (T )12 + C'#,T e6GoT
y·1 - y ' g(T.) 12C2
0

< e6GoT --.C ('Y) + --f + _2 -e6GoT

(

1 nCo ) C^2 T

• 2VT 3 12c5

< e6G^0 T --f + --d2 _ (p q) + --f + _2 -e6GoT

(

nCo e^2007 nCo ) C^2 T

• 3 4f g(T) ' 3 12c5

(

d;(-r) (p, q)) G

:::; 1 + - 2,

T

where C 2 is a constant depending only on n, To, T, and Co.
Hence, from (7.63), we have

IH (X, Y)I (T) :'.S C1 IY (T)l^2 +Co IY (T)l^2

T

+Co~ ( 1+ <lier)

7

(p, q)) C2 IY ( 7)1^2

+ c, H 1 + <lier)

7

(p, q)) c, · IY (7)1^2

:::; C2 (i + .!. + d;(-r) ~' q)) IY (T)l^2.

T TT