CHAPTER 19
Chapter 19. Geometric Properties of /\;-Solutions
Will the wind ever remember the names it has blown in the past?- From "The Wind Cries Mary" by Jimi Hendrix
In this chapter we discuss some basic properties of r;;-solutions. Topics
include the example of a nontrivial r;;-solution on the n-sphere, a generaliza-
tion of the notion of r;;-solution, the existence of an asymptotic shrinker in
a r;;-solution, and the r;;-gap theorem.
In §1 we review Perelman's no local collapsing theorem with applica-
tion to the existence of singularity models and we introduce the notion of
r;;-solution. We also collect some examples which provide some intuition re-
garding ancient solutions. For example, we give an elementary discussion of
the existence of c--necks in the Bryant soliton.
In §2 we modify Perelman's notion of r;;-solution by replacing the bounded
curvature assumption in his definition by the a priori weaker requirement
that the solution satisfies the trace Harnack estimate. Such solutions are
called 'r;;-solutions with Harnack'. We also introduce two notions reflecting
the geometry at infinity.
In §3 we present the construction of Perelman's r;;-solution on the n-
sphere sn for n 2: 3, which is rotationally symmetric, is invariant under a
reflection, and has the Bryant soliton as a backward limit and the shrink-
ing cylinder as another backward limit. Although Perelman's solution is
analogous to the 2-dimensional King-Rosenau solution (discussed in both
Volume One and Part I),^1 it is qualitatively different. In particular, Perel-
man's solution is r;;-noncollapsed at all scales for some r;; > 0, whereas the
King-Rosenau solution is not. It is possible (not proven) that Perelman's
solution is unique among Type II r;;-solutions on S^3.
In §4 we first recall Hamilton's classification of 2-dimensional r;;-solutions;
namely, the only such solutions are the round shrinking 2-sphere or its Z2-
quotient. In dimensions 2 and 3 we show the result that r;;-solutions with
Harnack must have bounded curvature and hence are r;;-solutions.
In §5 we discuss additional details (supplementing the proof of Theorem
8.32 in Part I) regarding the proof of the existence of an asymptotic gradient
shrinking soliton in a r;;-solution.
(^1) Perelman's solution and the King-Rosenau solution share the following characteris-
tics. They are rotationally symmetric, invariant under reflection, shrink to a round point
forward in time, and limit to either a cylinder or a steady soliton backward in time.
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