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 lim τ→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 R km i Rjkm, and the functional M |Rm|2 e−fdµ. 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
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