By Huai-Dong Cao, Xi-Ping Zhu.

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**Extra resources for A complete proof of the Poincare and geometrization conjectures - application of the Hamilton-Perelman theory of the Ricci flow**

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Hence d dt n M n (4πτ )− 2 e−f dV = −(4πτ )− 2 ∆(e−f )dV = 0. 12) 1 (4πτ )n/2 e−f dV = 1 M and ν(gij ) = inf W(g, f, τ ) | f ∈ C ∞ (M ), τ > 0, 1 (4πτ )n/2 e−f dV = 1 . Note that if we let u = e−f /2 , then the functional W can be expressed as W(gij , f, τ ) = n M [τ (Ru2 + 4|∇u|2 ) − u2 log u2 − nu2 ](4πτ )− 2 dV n n and the constraint M (4πτ )− 2 e−f dV = 1 becomes M u2 (4πτ )− 2 dV = 1. Thus µ(gij , τ ) corresponds to the best constant of a logarithmic Sobolev inequality. Since the nonquadratic term is subcritical (in view of Sobolev exponent), it is rather straightforward to show that inf n n [τ (4|∇u|2 + Ru2 ) − u2 log u2 − nu2 ](4πτ )− 2 dV M u2 (4πτ )− 2 dV = 1 M is achieved by some nonnegative function u ∈ H 1 (M ) which satisfies the EulerLagrange equation τ (−4∆u + Ru) − 2u log u − nu = µ(gij , τ )u.

Now we consider the Ricci flow on a Riemann surface. 5) becomes ∂gij = −Rgij . 3) on a Riemann surface M and 0 ≤ t < T . Then the scalar curvature R(x, t) evolves by the semilinear equation ∂R = △R + R2 ∂t on M × [0, 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.

The sup is taken over a compact subset of Rn × Rn . 3 d s(ϕ) ≤ sup{l(N (ϕ, t)) | η ∈ ∂Z, l ∈ Sη Z and s(ϕ) = l(ϕ − η)}. dt It is clear that the sup on the RHS of the above inequality can be takeen only when η is the unique closest point in Z to ϕ and l is the linear function of length one with gradient in the direction of ϕ − η. Since N (ϕ, t) is Lipschitz in ϕ and continuous in t, we have |N (ϕ, t) − N (η, t)| ≤ C|ϕ − η| for some constant C and all ϕ and η in the compact set Z. By hypothesis (ii), l(N (η, t)) ≤ 0, and for the unique η, the closest point in Z to ϕ, |ϕ − η| = s(ϕ).