Chapter 28. Special Ancient Solutions
When you're ridin' sixteen hours and the re's nothin' 1nuch to do
A n d you don't feel much like ridin', you just wish the trip was through.
- From "Turn the Page" by Bob Seger
The ancient solutions arising as singularity models (associated to finite-time
solutions of the Ricc i fl.ow on closed manifolds) a re necessarily 11:-noncoll apse d. In
P art III our disc ussion of an cient solut ions was primarily focused on P erelman's
theory of 3-dimensional ancient 11:-solutions. This theory is partly based on the
reduced volume monotonicity formula and on compactness arguments by subtle
point picking. On the other h and, in the physics literature, a ncient solutions with-
out the 11:-noncoll apsed condition are also co nsidered. This p artially motivates us
to consider general ancient solutions in t his ch apter.
In §1, we discuss a local estimate for the scalar curvature. A consequ ence of
t his estimate is that any complete ancient solution must h ave nonnegative scalar
curvature.
In §2, after mentioning so me properties and conjectures about singularity mod-
els which are reminiscent of t hose for gradient Ricci solitons (GRS), we prove some
results about 3-dimensional singularity models which were not disc ussed in the
earlier parts of this volume.
In §3 we discuss the classification of noncompact 2-dimensional ancient so lu-
tions.
In §4 we discuss some positive curvature conditions on ancient solutions which
imply that they a re shrinking spherical space forms.
1. Local estimate for the scalar curvature under Ricci flow
A priori, noncompact singularity models in dimensions at least 4 m ay not have
bounded curvature. The following result of B .-1. Chen implies that si ngularity
models always h ave nonnegative scalar curvature. The idea of its proof is to localize
the estimate that if Rmin (0) < 0 on a closed manifold, then
1
R > i.
THEOREM 28.l (Sharp lower bound for Ron complete solutions). If (Mn, g (t)),
t E (a, w), is a complete solution to the Ricci flow, then
n
(28.1) R > - on M x (a,w).
- 2 (t - a)
PROOF. Without loss of generality, we may assume that the so lution is defined
on the time interval [a, w).
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