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
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
and the functional
Throughout this chapter we assume that
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.
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