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By Menzies A.W.C., Lacoss D.A.

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Near Thus, the L-length of γ(q,¯τ ) varies smoothly with (q, τ¯), and defines a function L(q, ˆ = L near (p, 0). For if α : [0, τ¯] → M is a smooth curve then (p, 0). 6). Thus γ(q,¯τ ) minimizes when (q, τ¯) is close to (p, 0). We can now deduce that for all (q, τ¯), there is an L-geodesic γ : [0, τ¯] → M which has infimal L-length among all piecewise smooth curves starting at p and ending at q (with domain [0, τ¯]). This can be done by imitating the usual broken geodesic argument, using the fact that for x, y in a given small ball of M and for sufficiently small time intervals τ ′ +ǫ √ [τ ′ , τ ′ + ǫ] ⊂ [0, τ ], there is a unique minimizer γ for τ ′ τ R(γ(τ )) + |γ(τ ˙ )|2 dτ with NOTES ON PERELMAN’S PAPERS γ(τ ′ ) = x and γ(τ ′ + ǫ) = y.

6) to get a negative upper bound on M hv dV at time t = 0. Finally, M v dV is the expression that appears in the logarithmic Sobolev inequality. If the isoperimetric constant is sufficiently close to the Euclidean value cn then one concludes 54 BRUCE KLEINER AND JOHN LOTT that M hv dV must be bounded below by a constant close to zero, which contradicts the negative upper bound on M hv dV . 30. 1 We can assume that r0 = 1 and α < −1 αt }. 1 . 1 −1 does not hold then there is another point (x, t) with | Rm(x, t)| large (relative to t ) so that any other such point (x′ , t′ ) either has t′ > t or is much farther from x0 than x is.

From the evolution equation, dg∞ 1 g∞ . 4) = − 2 Ric(g∞ ) = 2 Hess f∞ − dt t−t It follows that the sectional curvatures of g∞ (x, t) go like (t∞ − t)−1 (after performing diffeomorphisms), which contradicts the fact that | Rm |(x∞ , t∞ ) = 1. Suppose now that there is a subsequence so that the injectivity radii of the scaled metrics at (xk , tk ) tend to zero. Parabolically rescale (Mk , (xk , tk ), gk (·)) further so that the injectivity radius becomes one. After passing to a subsequence we will have convergence to a flat Ricci flow solution (−∞, 0] × L.

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