1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

4. MORE RICCI FLOW THEORY AND ANCIENT SOLUTIONS 473

Theorem A.58 says the following. (See Chapter 6 for a definition of
r;;-noncollapsed.)
COROLLARY A.60 (Dimension reduction of steady solitons). If ( Mn,g ,J),
n 2: 3, is a complete steady gradient Ricci soliton which is r;;-noncollapsed

on all scales for some r;; > 0 and if sect (g) 2: 0, Re (g) > 0, and if R (g)

attains its maximum at some point, then a dilation about a sequence of points

tending to spatial infinity at time t = 0 converges to a complete solution

(M~, g 00 (t), x 00 ) which is the product of an (n - l)-dimensional solution^5

withR

Proposition 9.46 on p. 356 of [111]:

PROPOSITION A.61 (Re> 0 expanders have AVR > 0). If (Mn, g (t)),

t > 0, is a complete noncompact expanding gradient Ricci soliton with Re >

0, then AVR (g (t)) > 0.

Theorem 9.56 on p. 362 of [111]:

THEOREM A.62 (Steady or expander with pinched Ricci has R exponen-
tial decay). If (Mn, g) is a gradient Ricci soliton on a noncom pact manifold

with pinched Ricci curvature in the sense that Rij 2: ERgij for some E: > 0,

where R 2: 0, then the scalar curvature R has exponential decay.

Theorem 9.79 on pp. 375-376 of [111]:

THEOREM A.63 (Classification of 3-dimensional gradient shrinking soli-
tons with Rm 2: 0). In dimension 3, any nonflat complete shrinking gradient
Ricci soliton with bounded nonnegative sectional curvature is either a quo-
tient of the 3-sphere or a quotient of 82 x R

4.4. Ancient solutions. Theorem 10.48 on p. 417 of [111]:

THEOREM A.64 (Ancient solution with Rm 2: 0 and attaining sup R is
steady gradient soliton). If (Mn, g (t)), t E (-oo, w), is a complete solution
to the Ricci flow with nonnegative curvature operator, positive Ricci curva-
ture, and such that SUPMx(-oo,w) R is attained at some point in space and
time, then (Mn, g (t)) is a steady gradient Ricci soliton.

Analogous to the above result is the following:

THEOREM A.65 (Immortal solution with Rm 2: 0 and attaining sup tR

is gradient expander). If(Mn,g(t)), t E (0,oo), is a complete solution to

the Ricci flow with nonnegative curvature operator, positive Ricci curvature,

and such that SUPMx(O,oo) tR is attained at some point in space and time,

then (Mn, g ( t)) is an expanding gradient Ricci soliton.

Proposition 9.29 on p. 344 of [111]:

(^5) With bounded nonnegative sectional curvature.