# Huai-Dong Cao, Xi-Ping Zhu.'s A complete proof of the Poincare and geometrization PDF By Huai-Dong Cao, Xi-Ping Zhu.

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Case (ii): µ < 0. We need to verify dλ d(−µ) d(−ν) ≥ + (log(−ν) − 1) dt dt dt when λ = (−µ) + (−ν)[log(−ν) − 2]. -D. -P. ZHU which reduces to λ2 (−ν) + λ(−µ)2 + (−µ)2 (−ν) + (−ν)3 ≥ λ2 (−µ) + λ(−µ)(−ν) or equivalently (λ2 − λ(−µ) + (−µ)2 )((−ν) − (−µ)) + (−µ)3 + (−ν)3 ≥ 0. Since λ2 − λ(−µ) + (−µ)2 ≥ 0 and (−ν) − (−µ) ≥ 0 we are also done in the second case. Therefore the proof is completed. 5. Li-Yau-Hamilton Estimates. In , Li-Yau developed a fundamental gradient estimate, now called Li-Yau estimate, for positive solutions to the heat equation on a complete Riemannian manifold with nonnegative Ricci curvature.

T∈[0,T ] 223 THE HAMILTON-PERELMAN THEORY OF RICCI FLOW Suppose that, for any x ∈ M and any initial time t0 ∈ [0, T ), and for any solution σx (t) of the (ODE) which starts in Kx (t0 ), the solution σx (t) will remain in Kx (t) for all later times. Then for any initial time t0 ∈ [0, T ) the solution σ(x, t) of the (PDE) will remain in K(t) for all later times if σ(x, t) starts in K(t0 ) at time t0 and the solution σ(x, t) is uniformly bounded with respect to the bundle metric hab on M × [t0 , T ].

Suppose the scalar curvature of the initial metric is bounded, nonnegative everywhere and positive somewhere. 2 that the scalar curvature R(x, t) of the evolving metric remains nonnegative. Moreover, from the standard strong maximum principle (which works in each local coordinate neighborhood), the scalar curvature is positive everywhere for t > 0. In , Hamilton obtained the following Li-Yau estimate for the scalar curvature R(x, t). 2 (Hamilton ). Let gij (x, t) be a complete solution of the Ricci flow on a surface M .