1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

1. METRIC SPACES AND LENGTH SPACES

(2) If f: (X, dx) --+ (Y, dy) is an E-isometry, then

daH ((X, dx), (Y, dy )) < 2c.

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REMARK G.13. There is an exact relation between the Gromov-Haus-

dorff distance and the distortion of correspondences (see Theorem 7.3.25

of [18]; the definition of 'correspondence' is on p. 256 of [18]).

We say that a sequence of metric spaces {(Xi, dxi)}iEN converges in

the Gromov-Hausdorff distance to a metric space (Y, dy) if

i--+oo _lim daH ((Xi,dxJ, (Y,dy)) = 0.

1.2.2. Gromov-Hausdorff distance between pointed metric spaces.

When the metric spaces under discussion may be unbounded, we con-

sider pointed metric spaces obtained by adding a basepoint to the metric

space and we modify the definition of distance between them.^8 A (pointed)

map f : (X, xo) --+ (Y, Yo) is a map f : X--+ Y with f (xo) =YO·

The following is Definition 1.6 in [63].

DEFINITION G.14 (Approximate pointed isometries). Let (X,dx,xo)

and (Y, dy, Yo) be two pointed metric spaces. A map f : (X, xo) --+ (Y, yo)

is called an E-approximate pointed isometry (or E-pointed Hausdorff

approximation) if

(1)

(G.8)

(2)

(G.9)

for all x1,x2 E Bx (xo,E-^1 ).

The pointed Gromov-Hausdorff distance

d~H ((X, dx, xo), (Y, dy, Yo))

between two pointed metric spaces is defined to be the infimum over all

E > 0 such that there exist E-approximate pointed isometries from (X, xo)

to (Y, Yo) and from (Y, Yo) to (X, xo).^9

Note that for any x E Bx (xo,E-^1 ) we have

ldy (! (x), Yo) - dx (x, xo)I < E.

This implies that

(^8) We are interested in pointed spaces because solutions to the Ricci flow on noncompact
manifolds arise in the singularity theory of solutions of the Ricci flow on closed manifolds.
(^9) Note that when both (X, dx) and (Y, dy, yo) have bounded diameter, the pointed
Gromov-Hausdorff distance does not necessarily agree with the Gromov-Hausdorff
distance.