By Huai-Dong Cao, Xi-Ping Zhu.

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Additional info for A complete proof of the Poincare and geometrization conjectures - application of the Hamilton-Perelman theory of the Ricci flow

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THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 197 Here and in the following we denote by C various positive constants depending only on Ck and the dimension. Define Sk = Bk M 2 1 1 + k + Mk 2k r t + |∇k Rm|2 · |∇k+1 Rm|2 where Bk is a positive constant to be determined. By choosing Bk large enough and Cauchy inequality, we have ∂ Sk ≤ ∂t k + ∆|∇k Rm|2 − 2|∇k+1 Rm|2 tk+1 1 1 + CM 3 + k + Mk · |∇k+1 Rm|2 r2k t 1 1 + Bk M 2 + k + M k + |∇k Rm|2 r2k t − · ∆|∇k+1 Rm|2 − 2|∇k+2 Rm|2 + CM |∇k+1 Rm|2 + CM 3 1 1 + k+1 + M k+1 t r2(k+1) ≤ ∆Sk + 8|∇k Rm| · |∇k+1 Rm|2 · |∇k+2 Rm| − − 2|∇k+1 Rm|4 + CM 3 |∇k+1 Rm|2 k tk+1 |∇k+1 Rm|2 1 1 + k + Mk r2k t 1 1 + k + M k + |∇k Rm|2 r2k t 1 1 + CM |∇k+1 Rm|2 Bk M 2 + k + M k + |∇k Rm|2 r2k t 1 1 + CM 3 + k+1 + M k+1 t r2(k+1) 1 1 · Bk M 2 + k + M k + |∇k Rm|2 r2k t 1 1 ≤ ∆Sk − |∇k+1 Rm|4 + CBk2 M 6 + 2k + M 2k r4k t 1 1 + 2k+1 + M 2k+1 + CBk M 5 t r2(2k+1) 1 1 ≤ ∆Sk − |∇k+1 Rm|4 + CBk2 M 5 + 2k+1 + M 2k+1 t r2(2k+1) Sk ≤ ∆Sk − 2 (B + 1)2 M 4 r12k + t1k + M k − 2|∇k+2 Rm|2 Bk M 2 + CBk2 M 5 1 r2(2k+1) + 1 t2k+1 + M 2k+1 .

8) d¯ λ(gij (t)) dt ≥ 2V 2 n 1 + n 1 (R + ∆f )gij |2 e−f dV n M |Rij + ∇i ∇j f − M (R + ∆f )2 e−f dV − 2 (R + ∆f )e−f dV M ≥0 by the Cauchy-Schwarz inequality. Thus we have proved statement (i). We note that on an expanding breather on [t1 , t2 ] with αgij (t1 ) and gij (t2 ) differ only by a diffeomorphism for some α > 1, it would necessary have dV > 0, for some t ∈ [t1 , t2 ]. dt THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 205 On the other hand, for every t, − d 1 log V = dt V M RdV ≥ λ(gij (t)) by the definition of λ(gij (t)).

This pinching estimate plays a crucial role in analyzing the formation of singularities in the Ricci flow on three-manifolds. Consider a complete solution to the Ricci flow ∂ gij = −2Rij ∂t on a complete three-manifold with bounded curvature in space for each time t ≥ 0. 1) ∂ # 2 Mαβ = ∆Mαβ + Mαβ + Mαβ ∂t 2 where Mαβ is the operator square 2 Mαβ = Mαγ Mβγ # and Mαβ is the Lie algebra so(n) square # Mαβ = Cαγζ Cβηθ Mγη Mζθ . # In dimension n = 3, we know that Mαβ is the adjoint matrix of Mαβ . If we diagonalize Mαβ with eigenvalues λ ≥ µ ≥ ν so that   λ , µ (Mαβ ) =  ν # 2 then Mαβ and Mαβ are also diagonal, with  2   λ µν 2  and (M # ) =  µ2 (Mαβ )= αβ ν2 λν λµ  .

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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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