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Having completed a heuristic derivation of the monotonicity of Perelman reduced volume (Conjecture 1 from the previous lecture), we now turn to a rigorous proof. Whereas in the previous lecture we derived this monotonicity by converting a parabolic spacetime to a high-dimensional Riemannian manifold, and then formally applying tools such as the Bishop-Gromov inequality to that setting, our approach here shall take the opposite tack, finding parabolic analogues of the proof of the elliptic Bishop-Gromov inequality, in particular obtaining analogues of the classical first and second variation formulae for geodesics, in which the notion of length is replaced by the notion of ${\mathcal L}$-length introduced in the previous lecture.

The material here is primarily based on Perelman’s first paper and Müller’s book, but detailed treatments also appear in the paper of Ye, the notes of Kleiner-Lott, the book of Morgan-Tian, and the paper of Cao-Zhu.

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