CHAPTER 2
Special and limit solutions
In this chapter, we continue our study of special and intuitive solutions
to the Ricci flow. We introduce self-similar solutions, often called Ricci soli-
tons, which may be regarded as generalized fixed points of the flow. We
then give examples of solutions that exist for infinite time: eternal solutions
(those existing for all times -oo < t < oo), ancient solutions (those which
exist for times -oo < t < w), and immortal solutions (those which exist
for a < t < oo). Because these solutions have infinite time to diffuse, they
should have very special properties. Our interest in those properties arises
for the following reason. A singularity model for the Ricci flow is a com-
plete nonfiat solution obtained as a limit of dilations about a singularity. (In
Chapters 8 and 9, we shall study singularity models and their formation.)
Every singularity model exists for infinite time, and the special properties
one expects to find in a singularity model should yield valuable information
about the geometry of the original solution near the singularity just prior
to its formation. In this chapter, we also study two important singularities
directly: we present a rigorous analysis of neckpinch singularities (under
certain symmetry hypotheses) and a heuristic analysis of a degenerate neck-
pinch.
- Generalized fixed points
There is only a small class of genuine fixed points of the Ricci flow. A
Riemannian manifold (Mn, g) is a fixed point of the unnormalized Ricci flow
a
(2.1) otg = -2 Re
if and only if the metric g is Ricci fiat. A compact manifold (Mn, g) is a
fixed point of the normalized Ricci flow-g o = -2Rc+-^2 (f Mn Rdμ) g
ot n f Mn dμ
if and only if its average scalar curvature is constant (JMn Rdμ/ fMn dμ) =
p and
Re= £.g,
n
hence if and only if g is an Einstein metric.
There is however a larger class of solutions which may be regarded as
generalized fixed points. These are called self-similar solutions or Ricci
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