1547845440-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_III__Chow_

128 20. COMPACTNESS OF THE SPACE OF Ii-SOLUTIONS

Recall that (Ni (b,B) ,gi (0)) C (M,gi (0)) and note that

d 9 i(o) (p, xi) = V R(xi, to) · d;(to) (xi,P) :S: 2y/ ASCR(to)

for i sufficiently large. It follows easily from the uniqueness of the above

pointed Gromov-Hausdorff limit that if we change the basepoint p to a

sequence of basepoints Yi, a uniformly bounded distance away, then a corre-
sponding Gromov-Hausdorff limit is isometric to the original limit. Because

of this, as a metric space, the limit (N 00 (b,B) ,g 00 (0)) of the sequence

of pointed Riemannian manifolds {(Ni (b, B), 9i (0), Xi)} is isometric to an

open (top-dimensional) submanifold of the limit (CW, doo,Poo) of the se-

quence { ( M, 9i ( 0) , p)}. Hence W has a smooth ( n - 1 )-manifold structure

and there exists a Riemannian metric gw ( 0) on W such that the Euclidean
metric cone structure (CW, d 00 ) is given by the Riemannian metric

(20.3)

The Riemann curvature tensor Rm 900 (o) of the metric g 00 (0) = dr^2 +

r^2 gw (0) on N 00 (b, B) satisfies^6

(20.4) ( Rmg (^00) (o) ( :r' a~j)
0 ~j ' :r) = O,
where {yj} 7,:~ are local coordinates on W. Hence for each 1 :S: j :S: n - 1,

the 2-form gr/\ 8 ~j lies in the null space of the curvature operator of g 00 (0).

Recall that Hamilton's strong maximum principle for Rm (i.e., Theorem

12.50 and Theorem 12.53 in Part II) says that the null space of Rm 00 (0) is

invariant under parallel translation. Let V denote the covariant derivative

of 900(0). From^7

a a a _ 1 a

V EJ -;:;----= O and V Q -;:;----= V _§__ ~ = r -

8

8r ur EJyi ur ar uy^2 yi. ,

we have that the (infinitesimal) parallel translates of gr /\ 8 ~j, for 1 :S:

j :S: n - 1, span all of A^2 N 00 (b, B). Indeed, in local coordinates where

(^6) See, for example, equation (20.56) in the notes and commentary at the end of this
chapter or Proposition 9.106 of [15].
(^7) Let r = y (^0) so that 9oo(O)oo = 1, 9oo(O)ij = r^2 gw (O)ij' and 9oo(O)i0 = 0 for i,j?: l.
We have
[) n-1 k [) 0 [)
\7 §,_ By' <> ur = L::rio~ k=l uy +rio<> ur
1 k£ [). [) -1 [)
= 29 8r9i£ [)yk = r [)yi.

Note also that \7 8 ~, gr = \7 Ir 8 ~i since [ EJ~i, gr J = 0.