1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

378 26. BOUNDS FOR THE HEAT KERNEL FOR EVOLVING METRICS

PROPOSITION 26.49 (Distance-like functions with bounds on deriva-
tives). Given n E N - { 1} and K E ( 0, oo), there exists a constant Cn,K E

( 1, oo), depending only on n and K, such that if (Mn, g) is a complete non-

com pact Riemannian manifold with sectional curvature [sect (g) I :::; K and

if 0 E M, then there exists a C^00 function f : M -+ JR with the fallowing

bounds on M:

( 1) (distance-like)

(26.128) r (x) + 1:::; f (x) :::; r (x) + Cn,K,

where r (x) ~ d (x, 0), and

(2) (uniform bounds on first two derivatives)

(26.129) [\7 fl (x):::; Cn,K and [\7\7 fl (x) :::; Cn,K·

REMARK 26.50. There exist constants Cn,K,m E (O,oo), depending only

on n, K, and m, such that if we further assume that (Mn,g) satisfies

IV'kRml:::; K for 0:::; k:::; m, where m EN, then we have

(26.130)

for 1 :::; k :::; m + 2.

We now give the proof of this proposition, closely following Tam's paper

[177]. The idea is to start with a distance-like function with uniformly

bounded gradient and to evolve this function by the heat equation. The

evolved function retains the original properties of the initial function and

obtains the additional property of having uniformly bounded Hessian (and

even bounded higher derivatives if bounds on the derivatives of the curvature

are assumed).

By Proposition 2.1 in Greene and Wu [75], given (Mn, g) and 0 E M,

there exists a C^00 function u : M -+ JR with

(26.131) fu (x) - r (x)[ :::; 1 and f\7uf (x) :::; 2

on M (see also Lemma 12.30 in Part II).

Let H: M x M x (0, oo)-+ (0, oo) be the heat kernel (minimal positive

fundamental solution of the heat equation) of (M,g). Then the function

f : M x (0, oo) -+JR defined by

(26.132) f (x, t) ~JM H (x, y, t) u (y) dμ (y)

is a solution to the heat equation with limt-+O f (x, t) = u (x) uniformly in

xEM.

REMARK 26.51. The uniform convergence may be seen from the follow-

ing estimates which we shall prove below: (26.134), (26.139), and the bound

corresponding to (26.141), where the integration is over M - B (x, 6) for

any c5 E (0, 1], together with Remark 26.52.