- PERELMAN'S FORMALISM IN POTENTIALLY INFINITE DIMENSIONS 417
PROOF. The sequence ( M^3 ' 9,,.i ( e)' q,,.J' e ::::: 1, is a sequence of ancient
11;-solutions with bounded curvature for each i. The convergence 9,,.i ( B) -t
900 ( B) for B E [ A-1, A] implies that R 9 Ti ( q,,.i, 1) -t R 9 = (q 00 , 1). Since thelimit 900 (0) is nonfl.at, by the strong maximum principle, R 9 =(l) (q=) > 0.
By the compactness theorem of the set of 3-dimensional ancient 11;-solutions
(see Part II of this volume), we know that a subsequence(M, R;~ (l,q-ri)9,,.J B), q,,.i)
converges to a limit (M 00 , g 00 (B), q 00 ). The limit has bounded sectional
curvature. Using the definition of Cheeger-Gromov convergence, it is easy to
check that (M, R;~(l,q=)9,,.i (B), q~i) converges to the limit (M 00 , g 00 (B), q 00 ).
On the other hand it follows from Theorem 8.32 that(M, R;~(l,q<XJ)9Ti (B), q,,.i) -7 (Moo, R;~(l,q<XJ)9oo(B), qoo).
Hence, by the uniqueness of the Cheeger-Gromov pointed limit, 900 (0) =
R 9 =(l,q=)g 00 (B) for (} E [1, A], which has bounded curvature. We have
proved that the shrinking gradient soliton 900 ( B) is complete and has nonfl.at
bounded nonnegative curvature. D- Perelman's Riemannian formalism in potentially infinite
dimensions
Here we discuss Perelman's potentially infinite Riemannian metric in
more detail. We discuss the calculation of its curvature tensor, the Ricci
flatness (modulo renormalization) of its metric, and a geometric interpreta-
tion of Perelman's entropy formula. Although this section does not belong
to the linear fl.ow of this chapter, it provides a unifying viewpoint for various
components of this volume. In particular, we have the following.
(1) In Section 2.1 of Chapter 7 we considered the Riemannian metric
g on M =Mn x SN x (0, T) and showed that the renormalization
of its Riemannian length is the £-length.^7
(2) In subsection 3.3 of Chapter 7 we showed that the £-geodesic equa-
tion is the same (up to the time reparametrization O" = 2./T) as the
geodesic equation of the space-time connection defined by (7.39)-(7.42), which is the limit, as N -too, of the Levi-Civita connections
of the Riemannian metrics g defined by (7.11) (see Exercise 7.4).
Thus the £-geodesic equation is the limit, as N -t oo, of the geo-
desic equations of the Riemannian metrics g.
(3) In subsection 2.1 of this chapter we showed how to obtain the re-
duced volume functional from a renormalizatio.£._ of the volumes of
geodesic balls with respect to the metric g on M.(^7) Here we have switched the notation from h tog and N to M.