CHAPTER 17

Entropy, µ-invariant, and Finite Time Singularities

I’ll tip my hat to the new constitution.

– From “Won’t Get Fooled Again” by The Who

Monotonicity formulas may be used to understand the qualitative be-

havior of solutions of the Ricci flow. As an example, in this chapter we

consider the µ-invariant monotonicity formula and its applications to singu-

larity analysis.

In §1 we discuss lower and upper bounds for the µ-invariant. As an

application, there is a lower bound for the volume of solutions g (t) of the

Ricci flow with nonpositive λ-invariant. This implies that the corresponding

finite time singularity models are noncompact in this case. As a further

application, we discuss the classification of compact finite time singularity

models as shrinking gradient Ricci solitons with no assumption on the sign

of the λ-invariant.

In §2 we prove the fact that limτ→0+ µ (g, τ) = 0. This result was stated

as Lemma 6.33(ii) in Part I but was not proved there. As an application we

show that, for a closed Riemannian manifold on which the isometry group

acts transitively, the minimizer for W is not unique for suﬃciently small τ.

In §3 we revisit the proof of the existence of a smooth minimizer for W

while completing some additional details not discussed earlier in this book

series.

One may hope to extend Perelman’s energy and entropy monotonic-

ity formulas. In §4 we discuss formulas relating Perelman’s energy F, the

linear trace Harnack quadratic, Hamilton’s matrix quadratic, the 2-tensor

RikmRjkm,

and the functional

M

|Rm|2

e−f

dµ.

Throughout this chapter we assume that

Mn

is a closed manifold unless

otherwise indicated.

1. Compact finite time singularity models are shrinkers

In this section and the next we discuss properties of the µ-invariant of a

metric g at a scale τ 0. This invariant is the infimum of Perelman’s entropy

functional W (g, f, τ), under a constraint, considered in Chapter 6 of Part I.

In this section we present the results and proofs of Z.-L. Zhang on bounds

for the µ-invariant and their geometric application to the classification of

compact finite time singularity models.

1

http://dx.doi.org/10.1090/surv/163/01