Chapter 6. Entropy and No Local Collapsing
Everything should be made as simple as possible, but not simpler.- Albert Einstein
Disorder increases with time because we measure time in the direction in which
disorder increases. -Stephen Hawking
Close, but no cigar. -Unknown origin
By combining Perelman's energy and the classical entropy in a suit-
able way, we obtain the entropy functional W, which we shall discuss in
this chapter. This is implemented with the introduction of a positive scale-
factor T. The advantage which the addition of this scale-factor yields is that
from the functional W we can understand aspects of the local geometry of
the manifold, e.g., volume ratios of balls with radius on the order of Vr·
Perelman's entropy is also the integral of his Harnack quantity for funda-
mental solutions of the adjoint heat equation.^1 As such, one can integrate
in space the Harnack partial differential inequality to give a proof of the
monotonicity formula for W. Note that here Perelman's Harnack quantity
is directly related to the entropy whereas in Hamilton's earlier work on sur-
faces, Hamilton's Harnack quantity is related to the time-derivative of his
entropy.^2 This monotonicity formula can be used to prove that shrinking
breathers must be shrinking gradient Ricci solitons. More importantly, this
monotonicity formula will be fundamental in proving Hamilton's little loop
conjecture or what Perelman calls the no local collapsing theorem.
In this chapter, we shall discuss in detail the entropy estimates, the two
functionals μ and v associated to W, and their geometric applications. We
discuss the logarithmic Sobolev inequality, which is related to the entropy
functional. We will also give different versions/proofs of the no local collaps-
ing theorem. In the last part of this chapter we shall discuss some interesting
calculations related to entropy.
Throughout this chapter Mn denotes a closed n-dimensional manifold.
1. The entropy functional W and its monotonicity
Let (Mn, g(t)), t E [O, T], be a solution of the Ricci flow on a closed
manifold. Note that by the proof of Lemma 5.31, we have that when
(^1) See (6.21).
(^2) Equation (6.21) as compared to (5.70).
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