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A complete proof of the Poincare and geometrization by Huai-Dong Cao, Xi-Ping Zhu.

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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By the above discussion, we know R(x, t) > 0 for t > 0. 4) is equivalent to ∂L 1 1 − |∇L|2 + = △L + R + ≥ 0. 5) Q= ∂L − |∇L|2 = △L + R. ∂t Then by a direct computation, ∂Q ∂ = (△L + R) ∂t ∂t ∂L ∂R =△ + R△L + ∂t ∂t = △Q + 2∇L · ∇Q + 2|∇2 L|2 + 2R(△L) + R2 ≥ △Q + 2∇L · ∇Q + Q2 . -D. -P. ZHU So we get ∂ ∂t Q+ 1 t ≥△ Q+ 1 t + 2∇L · ∇ Q + 1 t + Q− 1 t Q+ 1 t . 3, we obtain Q+ 1 ≥ 0. t This proves the theorem. As an immediate consequence, we obtain the following Harnack inequality for the scalar curvature R by taking the Li-Yau type path integral as in [82].

1) ∂u |∇u|2 n − + u≥0 ∂t u 2t on M × (0, ∞). 2) ∂u n + 2∇u · V + u|V |2 + u ≥ 0. 1). 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.

E. ∆ (∇t )i hab = (∇t ) ∂ ∂xi hab = 0, ∂ ∂ for any local coordinate { ∂x 1 , . . , ∂xn }. The Laplacian ∆t acting on a section σ ∈ Γ(V ) is defined by ∆t σ = g ij (x, t)(∇t )i (∇t )j σ. For the application to the Ricci flow, we will always assume that the metrics gij (·, t) evolve by the Ricci flow. Since M may be noncompact, we assume that, for the sake of simplicity, the curvature of gij (t) is uniformly bounded on M × [0, T ]. , N (x, σ, t) is a timedependent vector field defined on the bundle V and tangent to the fibers.

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