286 7. THE REDUCED DISTANCE/'\;-solutions, finite-time singularity models, and high curvature re-
gions of the solution.
( 5) Relating Ricci fl.ow and aspects of function theory and the heat
equation, for example, the analysis of the fundamental solution of
the adjoint heat equation coupled to the Ricci fl.ow.
In this chapter we discuss the basic properties of the £-length and the as-
sociated distance functions for complete (not necessarily compact) solutions
to the backward Ricci fl.ow with bounded sectional curvature.^1
1. The £-length and distance for a static metric
One of the primary antecedents of Perelman's £-length and reduced
distance f is the work of Li-Yau on differential Harnack inequalities for the
heat equation on a Riemannian manifold with a static metric.^2 With this
in mind we start by summarizing the properties of the energy functional for
paths in a Riemannian manifold and various monotonicity and comparison
results. The purpose for this is to compare properties associated to the linear
heat equation with respect to a static metric to properties of the nonlinear
case of metrics evolving by Ricci fl.ow and to show a strong analogy between
these two cases. The fact that the case of the heat equation is less technical
facilitates the presentation of some of the underlying ideas.
Let (Mn, g) be a complete Riemannian manifold. Given a C^1 -path 'Y :
[r1, r2] --+ M, T1 2: 0, joining two points, i.e., 'Y(r1) = p and 'Y(r2) = q, we
define its energy by
Convention: By saying that a path 'Y ( T) is ck' where k = 1 or 2,
we really mean 'Y ( ~2
) is a Ck function of O' ~ 2.jT.
This energy functional, which is well-suited for studying the heat equa-
tion, is equivalent to the usual energy for paths (see for example §12 of
Milnor's book [265]). Indeed, making the change of variables O' = 2../i
yields
1
&TI, T2 ('Y) = 2y'T2 I d d"'f^12 dO' •
2y'T1 (J' g(7.1)
Keeping in mind that we shall be discussing parabolic equations, we require
that the parameter of the path be given by the time variable T. In particular,
if we want to study the (reversed) concentration process of the fundamental
(^1) See the last section for some of the notational conventions we use in this chapter.
(^2) The later work of Hamilton on the matrix differential Harnack for the Ricci flow is
also important in this development.