Chapter 18. Geometric Tools and Point Picking Methods
Funny how my memory slips while looking over manuscripts of unpublished rhyme.- From "Hazy Shade of Winter" by Simon and Garfunkel
In this chapter we primarily discuss some tools used extensively in the
qualitative study of solutions of the Ricci fl.ow. Applications of these tools
appear in subsequent chapters and the impatient reader may wish to skip
this chapter while referring back to it when necessary.
A basic invariant reflecting the geometry of a Riemannian manifold is the
distance function. We begin the chapter by discussing in §1 the changing
distances estimate for solutions of Ricci fl.ow. The results in this section
are useful for a variety of applications: localizing monotonicity formulas,
controlling geometric invariants depending on the distance function such
as the diameter and the asymptotic scalar curvature ratio, and estimating
space-time geometric invariants such as the reduced distance.
To introduce the next topic, we recall the popular real estate saying
"location, location, location." This applies to Ricci fl.ow in the form of point
picking. Suppose that we are given a solution to the Ricci fl.ow, such as a
finite time singular solution to the Ricci fl.ow on a closed manifold or a Kr
solution. The purpose of point picking is to obtain a good sequence of points
and times (locations) so that the corresponding rescaled solutions based at
these points and times admit a subsequence with a limiting solution having
nice properties. We usually rescale so that the norms of the curvatures (or
the scalar curvatures) of the rescaled solutions equal 1 at the basepoints.
The reason for doing this is to obtain a nontrivial limit.
One of Perelman's innovations in [152] is the deep and clever use of
point picking. These methods are useful for obtaining crucial curvature
bounds via contradiction arguments and studying the geometry at infinity
of t>;-solutions (see Definition 19.7 in this volume).^1 In §2 and §3 we discuss
various point picking methods foundational to Perelman's work on Ricci
fl.ow. Point picking methods are further discussed in Chapter 22 related to
pseudo locality.
(^1) Results and tools which give strength to these arguments are the no local collapsing
theorem, volume comparison, ASCR = oo, AVR = 0, dimension reduction, classification
of 2-dimensional ancient K:-solutions, and the strong maximum principle.
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