1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

2. REDUCED VOLUME FOR RICCI FLOW 387

2.1. Volumes of geodesic spheres in M. We motivate the definition

of the reduced volume by computing the volume of geodesic spheres in the

potentially infinite-dimensional manifold (M, g) introduced in subsection

2.1 of Chapter 7. In particular, let p = (x 0 , yo, 0), f E (0, T), and

B9 (p, v2IVf) c M ~ M x SN x (0, T)

denote the ball centered at p with radius v'2l.Vf-with respect to the metric:

g ~ 9ijdxidxj + T9cx13dycxdyf3 + (~ + R) dT^2 ,

where 9cx/3 is the metric on SN of constant sectional curvature 1/ (2N).

For any point w = (x, y, Tw) E 8B9(p, v'2i.Vf-), because of the factor T in
T9cx13dycxdyf3, we have

v2iVf = d9 (w,p) = d9 ((x, y, Tw), (xo, Yo, 0))

= d9 ((x,y,Tw), (xo,y,0)).

Hence, letting/ (T) ~ (/M (T), y, T), TE [O, Tw], with/ (0) = (xo, y, 0) and

/M (Tw) = w, we have

(8.14)

where

v2iVf = inf Length g ( /)

'Y

=inf ( k J;w VT ( R + ['YM (T)[2) dT )

'YM +)2N T w + 0 ( N-^3 /^2 )

= yL;1v·1w+ ~ .J'iNL^1 ( x,Tw ) +O ( N -3/2) ,

L(x, Tw) ~ ~~ forw VT ( R + ['YM (7)[^2 ) dT.

and the infimum is taken over /M : [O, Tw] -+ M with /M (0) = xo and

/M ( Tw) = x. Therefore for any w = (x, y, Tw) E 8B9(p, '\/'2JVf),

yiT;; = Vf-

2

~L(x,Tw) + 0 (N-^2 ).

This implies that the geodesic sphere 8B9 (p, v'21.Vf-) , with respect tog, is

O(N-^1 )-close to the hypersurface M x SN x {f}. ·

EXERCISE 8.12. Justify the equality (8.14).

Note that since the fibers sN pinch to a point as T -+ 0, if w =

(x,y,Tw) E 8B9(p, v'2i.Vf-), then any point in {x} x SN x {Tw} also lies

on the sphere 8B 9 (p, v'21.Vf-). We have that the volume of 8B9 (p, v'21.Vf-)