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4) is preserved by the ODE. 5) whenever ν ≤ −e2 . λ + µ ≥ (−ν)[log(−ν) − 2], To show the inequality is preserved we only need to look at points on the boundary of the set. If ν + f −1 (λ + µ + ν) = 0 then ν = −f −1 (λ + µ + ν) ≤ −e2 since f −1 (y) ≥ e2 . 5) is nonnegative. We thus have λ ≥ 0 because λ ≥ µ. But µ may have either sign. We split our consideration into two cases: Case (i): µ ≥ 0. We need to verify dλ dµ d(−ν) + ≥ (log(−ν) − 1) dt dt dt when λ + µ = (−ν)[log(−ν) − 2]. Solving for log(−ν) − 2 = λ+µ (−ν) and substituting above, we must show λ2 + µν + µ2 + λν ≥ λ+µ + 1 (−ν 2 − λµ) (−ν) which is equivalent to (λ2 + µ2 )(−ν) + λµ(λ + µ + (−ν)) + (−ν)3 ≥ 0.

9) W(gij , f, τ ) = n M [τ (R + |∇f |2 ) + f − n](4πτ )− 2 e−f dV where gij is a Riemannian metric, f is a smooth function on M , and τ is a positive scale parameter. Clearly the functional W is invariant under simultaneous scaling of τ and gij (or equivalently the parabolic scaling), and invariant under diffeomorphism. Namely, for any positive number a and any diffeomorphism ϕ W(aϕ∗ gij , ϕ∗ f, aτ ) = W(gij , f, τ ). 2, we have the following first variation formula for W. 7 (Perelman [103]). If vij = δgij , h = δf, and η = δτ , then δW(vij , h, η) = M −τ vij Rij + ∇i ∇j f − 1 n gij (4πτ )− 2 e−f dV 2τ v n n −h− η [τ (R + 2∆f − |∇f |2 ) + f − n − 1](4πτ )− 2 e−f dV 2 2τ M n n + η R + |∇f |2 − (4πτ )− 2 e−f dV.

5), we have d s(t) ≤ Cs(t) + ǫ(∆t f (x) + ui (∇t )i f (x) + (C − A)f (x))eAt dt ≤ Cs(t) for A > 0 large enough, since f (x) ≥ 1 and the first and second covariant derivatives of f are uniformly bounded on M × [0, T ]. 1. 1 by Chow and Lu in [40] which allows the set K to depend on time. One can consult the paper [40] for the proof. 5 (Chow and Lu [40]). Let K(t) ⊂ V , t ∈ [0, T ] be closed subsets which satisfy the following hypotheses (H3) K(t) is invariant under parallel translation defined by the connection ∇t for each t ∈ [0, T ]; ∆ (H4) in each fiber Vx , the set Kx (t) = K(t) ∩ Vx is nonempty, closed and convex for each t ∈ [0, T ]; (H5) the space-time track (∂K(t) × {t}) is a closed subset of V × [0, T ].

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A complete proof of the Poincare and geometrization conjectures - application of the Hamilton-Perelman theory of the Ricci flow by Huai-Dong Cao, Xi-Ping Zhu.


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