By Huai-Dong Cao, Xi-Ping Zhu.

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4. (i) λ(gij (t)) is nondecreasing along the Ricci flow and the monotonicity is strict unless we are on a steady gradient soliton; (ii) A steady breather is necessarily a steady gradient soliton. To deal with the expanding case we consider a scale invariant version ¯ ij ) = λ(gij )V n2 (gij ). λ(g Here V = V ol(gij ) denotes the volume of M with respect to the metric gij . -D. -P. 5. ¯ ij ) is nondecreasing along the Ricci flow whenever it is nonpositive; more(i) λ(g over, the monotonicity is strict unless we are on a gradient expanding soliton; (ii) An expanding breather is necessarily an expanding gradient soliton.

D. -P. ZHU Set ϕ = rg s2 r2 , where s is the geodesic distance function from p with respect to the metric at t = 0. Then ∇ϕ = 1 ′ g r s2 r2 · 2s∇s and hence |∇ϕ| ≤ 2C1 . Also, ∇2 ϕ = 1 ′′ g r s2 r2 1 2 1 4s ∇s · ∇s + g ′ 2 r r s2 r2 1 2∇s · ∇s + g ′ r s2 r2 · 2s∇2 s. Thus, by using the standard Hessian comparison, C1 C1 + |s∇2 s| r r C1 C2 √ ≤ 1+s + M r s C3 ≤ . r |∇2 ϕ| ≤ Here C1 , C2 and C3 are positive constants depending only on the dimension. θ ] by letting ϕ to be zero outside B0 (p, r) and indepenNow extend ϕ to U × [0, M dent of time.

Then, at (x0 , t0 ), we have ∂ ˜ (Mαβ v α v β ) ≤ 0, ∂t ˜ αβ v α v β ) = 0, ∇(M ˜ αβ v α v β ) ≥ 0. and ∆(M THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 213 But 0≥ ∂ ˜ ∂ (Mαβ v α v β ) = (Mαβ v α v β + εeAt f ), ∂t ∂t ˜ αβ v α v β ) − ∆(εeAt f ) + ui ∇i (M ˜ αβ v α v β ) = ∆(M − ui ∇i (εeAt f ) + Nαβ v α v β + εAeAt0 f (x0 ) ≥ −CεeAt0 f (x0 ) + εAeAt0 f (x0 ) > 0 when A is chosen sufficiently large. This is a contradiction. 3 to the evolution equation ∂ # 2 Mαβ = ∆Mαβ + Mαβ + Mαβ ∂t of the curvature operator Mαβ , we immediately obtain the following important result.

### 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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