Chapter 7. The Reduced Distance
How thoroughly it is ingrained in mathematical science that every real advance
goes hand in hand with the invention of sharper tools and simpler methods
which, at the same time, assist in understanding earlier theories and in casting
aside some more complicated developments. -David HilbertTechnical skill is mastery of complexity while creativity is mastery of simplicity.- Chris Zeeman
In [297] Perelman introduced a new length (energy-like) functional for
paths in the space-times of solutions of the Ricci flow, called the £-length.
The naturalness of this functional can be justified both by the space-time ap-
proach and the various differential inequalities that the quantities associated
to the £-length satisfy, which we shall show in this chapter. A fundamental
inequality is the monotonicity of the reduced volume. As we shall see in
the next chapter, this monotonicity leads to a second proof of (weakened)
no local collapsing for finite time singularities. We emphasize that, unlike
the entropy proof, the proof of weakened no local collapsing in Chapter 8
using the reduced volume also holds for complete noncompact solutions with
bounded sectional curvature. Recall that no local collapsing provides a local
injectivity radius estimate and at the same time rules out the formation of
the cigar soliton singularity model.
Besides bringing comparison geometry and integral monotonicity into
Ricci flow, some original aspects of Perelman's work on the reduced distance
function are as follows. ·
(1) A space-time distance-like function which is not always nonnega-
tive.
(2) Bochner formulas, i.e., partial differential inequalities, which are
geometry (e.g., curvature) independent, i.e., these formulas and in-
equalities hold in any dimension and are independent of the initial
metric.
(3) Pointwise monotonicity adapted to the space-time geometry.· This
describes in some sense how, for any solution to the Ricci flow, the
geometry improves as time increases.
(4) Using the space-time geometry to understand point-picking and
compactness, in particular, understanding the· structure of ancient
285