1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

1. METRIC SPACES AND LENGTH SPACES 393

given by

L(1) ~L 9 (1) ~ 1bl~= (u),

9

du.

Let A denote the space of piecewise C^1 paths in M; then ( M, JR, A, L) is a

length space. The distance function is defined by

d (x, y) =inf L (r),

I

where the infinrnm is taken over all piecewise C^1 paths 1' joining x to y.

Note that the infimum is realized by at least one geodesic joining x and y

provided g is complete. We then have that (M, d) is a metric space.

1.1.4. Relevance to the Ricci flow.

A more sophisticated example related to length-type spaces, fundamen-

tal to the study of the Ricci fl.ow, is the reduced distance; see §7 of Perelman

[152] and Chapter 7 of Part I. Note that the £-length may be negative and

is not an example of a length function.

Another relevance of length spaces to Ricci fl.ow (and more generally,

Riemannian geometry) is via Gromov's compactness theorem, which we re-

call in the next subsection. Aleksandrov spaces (see the next section for a

definition), as limits of Riemannian manifolds, naturally occur in Perelman's

work on singularity analysis in Ricci fl.ow [152].

1.2. Gromov-Hausdorff distance and Gromov's precompact-

ness theorem.

In this subsection we review the definition of Gromov-Hausdorff ( GH)

distance between metric spaces and Gromov's precompactness theorem for

Riemannian manifolds with Ricci curvature bounded from below in the col-

lection of metric spaces.

1.2.1. Gromov-Hausdorff distance between metric spaces.

Given a metric space (Z, dz), the (closed) E-neighborhood of a subset

S is defined by

N 6 (S) ~ {z E Z : dz (z, S) :::; E}.

Given two subsets A and B of Z, the Hausdorff distance d~ (A, B) be-

tween them is defined to be the infimum over all E > 0 such that A is con-

tained in the E-neighborhood of Band Bis contained in the E-neighborhood

of A, that is,

d~ (A, B) ~ inf { E > 0 : A c N 6 ( B) and B c N 6 (A)}.

If there is no such E, then we defined~ (A, B) ~ oo. Note that d~ is a metric
on the set of compact subsets of Z. (See §3.3 of Gromov [78].)
Using the Hausdorff distance, we may define the Gromov-Hausdorff
distance dGH ( (X, dx) , (Y, dy)) between two metric spaces (X, dx) and