1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

2. ALEKSANDROV SPACES WITH· CURVATURE BOUNDED FROM BELOW 401

If we have a variation or = s, then

(G.15)

OL('Y) = !,"''"'') (r'+ (~:)'f

1

' (rs+~:~:) du

= !,"''"") (r' + (::) ')-s;z (-r ::: + 2 (::)

2

+ r^2 ) rsdu,

where we integrated by parts and used the boundary conditions s (0) =

s ( d ( x1, x2)) = 0. For any a, b E JR, the function

1

(G.16) r(u) = ----

acosu + bsinu

is a solution to the Euler-Lagrange equation of (G.15):

d

2

r (dr)

2

2

-r du2 + 2 du + r = 0.

Note that the form of (G.16) matches with (G.14).

EXERCISE G.32 (Riemannian cone is the metric cone). Show that if

(Mn, g) is a Riemannian manifold, then the distance function dgcone of the

Riemannian metric 9Cone = r^2 g + dr^2 on M x (0, oo) c Cone (M) is the

same as the cone metric dcone(M) on Cone (M) restricted to M x (0, oo )".

It is a straightforward calculation that, for n 2: 2, (Mn, g) has sectional

curvature bounded from below by 1 if and only if (M x (0, oo) , 9Cone) has

sectional curvature bounded from below by 0. This result has been general-

ized to complete length spaces (see Theorem 4.2.3 on p. 13 of [19]).

2. Aleksandrov spaces with curvature bounded from below

To define the curvature in the traditional way (as the Riemann curva-
ture tensor), a Riemannian metric must be 02. However, the subcollection
of complete Ck Riemannian manifolds, for any k 2: 2, even with both curva-
ture and diameter bounds, is not precompact in the collection of complete
02 Riemannian manifolds. This lack of precompactness is a motivation to
enlarge the collection of complete Riemannian manifolds to geometric spaces
with 'less regularity', in particular, including at least all the limiting metric
spaces. Note that one of the main issues in the failure of compactness is that
the injectivity radii of a sequence of such manifolds may not be bounded
from below by a positive constant.
In understanding the limiting behavior of sequences and families of Rie-
mannian metrics, important notions are that of Gromov-Hausdorff conver-
gence and compactness. As we have seen, these notions are defined for
spaces with less regularity and provide a framework to discuss Gromov's
precompactness theorem. Moreover, one of the main techniques to prove
precompactness (Theorem G.17) is the volume comparison theorem. In