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. 3,157 more words

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#### 285G, Lecture 9: Comparison geometry, the high-dimensional limit, and Perelman reduced volume

We now turn to Perelman’s second scale-invariant monotone quantity for Ricci flow, now known as the *Perelman reduced volume*. We saw in the previous lecture that the monotonicity for Perelman entropy was ultimately derived (after some twists and turns) from the monotonicity of a potential under gradient flow. 3,708 more words