1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

362 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS

for i = 1, 2. Using the semigroup property (26.41), which implies

we compute for t > 0

(26.79)

Now let

(26.80) t:.( )_,_d

(^2) (x,Ui)
<,,i x, t -;- 2t
for i = 1, 2 (note that ei = 0 in Ui), where d (x, Ui) ~ infyEUi d (x, y) denotes
the distance from x to Ui. Note that by the triangle inequality, d ( ·, Ui) is
a Lipschitz function with Lipschitz constant 1, so that
(26.81)
wherever d ( ·, Ui) is differentiable, which is a.e. on M by Rademacher's
theorem.
Since for any x E M

(26.82)

One calculates from (26.80) and (26.81) that

oei = _ d

2

(x, Ui) < _! IVt:·l2.

at 2t^2 - 2 <,,i