# 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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3) ∂ l(σ(x, t) − η) − ǫAeAt f (x) ∂t ∂σ(x, t) − ǫAeAt f (x) =l ∂t = l(∆t σ(x, t)) + l(ui (x, t)(∇t )i σ(x, t)) + l(N (x, σ(x, t), t)) − ǫAeAt f (x). 4 we have η + N (x, η, t) ∈ Tη Kx . -D. -P. ZHU for some positive constant C by the assumption that N (x, σ, t) is Lipschitz in σ and the fact that the sup is taken on a compact set. 5) ds(t) ≤ l(∆t σ(x, t)) + l(ui (x, t)(∇t )i σ(x, t)) + Cs(t) + ǫ(C − A)eAt f (x) dt for those x ∈ M, η ∈ ∂Kx and l ∈ Sη Kx such that l(σ(x, t) − η) − ǫeAt f (x) = s(t).

An important application of the advanced maximum principle is the Hamilton-Ivey curvature pinching estimate for the Ricci flow on three-manifolds given in the next section. More applications will be given in Chapter 5. Let M be a complete manifold equipped with a one-parameter family of Riemannian metrics gij (t), 0 ≤ t ≤ T , with T < +∞. Let V → M be a vector bundle with a time-independent bundle metric hab and Γ(V ) be the vector space of C ∞ sections of V . e. ∆ (∇t )i hab = (∇t ) ∂ ∂xi hab = 0, ∂ ∂ for any local coordinate { ∂x 1 , .

T ∂v α ∂t This shows that ∂v ∈ null (Mαβ ), ∂t so the null space of Mαβ is invariant in time. We now apply Hamilton’s strong maximum principle to the evolution equation of the curvature operator Mαβ . Recall ∂Mαβ # 2 = ∆Mαβ + Mαβ + Mαβ ∂t # where Mαβ = Cαξγ Cβηθ Mξη Mγθ . Suppose we have a solution to the Ricci flow with nonnegative curvature operator. 1, the null space of the curvature operator Mαβ of the solution has constant rank and is invariant in time and under parallel translation over some time interval 0 < t < δ .