1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

452 A. BASIC RICCI FLOW THEORY

then

inj (p) ;:::: lo.

1.11. Laplacian and Hessian comparison theorems. Given KE JR.

and r > 0, let

{

( n -1) VK cot ( VK r)

Hg (r) "c (n-1) ~~:th ( vlKfr)

if K > 0,

if K = 0,

if K < 0,

where if K > 0 we assumer < 2 :JK. The function HK (r) is equal to the

mean curvature of the ( n - 1 )-sphere of radius r in the complete simply-
connected Riemannian manifold of constant sectional curvature K.

THEOREM A.8 (Laplacian comparison). Let (Mn,g) be a complete Rie-

mannian manifold with Re ;:::: ( n - 1) K, where K E R For any p E Mn and

x E Mn at which dp ( x) is smooth, we have

(A.9) b.dp (x) ::; HJ{ (dp (x)).

On the whole manifold, the Laplacian comparison theorem (A.9) holds in

the sense of distributions. That is, for any nonnegative C^00 function <p

on Mn with compact support, we have

{ dp (x) b.c.p (x) dμ (x) :S: { CK (dp (x)) <p (x) dμ (x).

}Mn }Mn

The following is a special case of the Hessian comparison theorem.

THEOREM A.9 (Hessian comparison theorem). If (Mn, g) is a complete

Riemannian manifold with sect ;:::: K, then for any point p E M the distance
function satisfies

(A.10)

at all points where dp is smooth (i.e., away from p and the cut locus). On

all of M the above inequality holds in the sense of support functions.

That is, for every point x E M and unit tangent vector V E TxM, there

exists a C^2 function v: (-.s, .s)---+ JR. with v (0) = dp (x)'

dp (expx (tV)) :S: v (t), fort E (-.s, .s),

and

d

d: I v (t) ::; ~

1

HK (dp (x)).
t t=O n
Note that n~l HK (r) is equal to the principal curvature of the totally
umbillic ( n - 1 )-sphere of radius r in the complete simply-connected Rie-
mannian manifold of constant sectional curvature K. In fact b.dp is the
mean curvature of the distance sphere in M whereas V'V' dp is the second

fundamental form of the distance sphere in M.