New PDF release: A complete proof of the Poincare and geometrization

By Huai-Dong Cao, Xi-Ping Zhu.

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1) becomes an equality on our expanding soliton solution u! 8). Based on the above observation and using a similar process, Hamilton found a matrix quantity, which vanishes on expanding gradient Ricci solitons and is nonnegative for any solution to the Ricci flow with nonnegative curvature operator. Now we describe the process of finding the Li-Yau-Hamilton quadratic for the Ricci flow in arbitrary dimension. 3. Here Vb = ∇b f for some function f . 10) Rab + ∇a Rbc − ∇b Rac = ∇a ∇b Vc − ∇b ∇a Vc = Rabcd Vd , and differentiating again, we get ∇a ∇b Rcd − ∇a ∇c Rbd = ∇a (Rbcde Ve ) = ∇a Rbcde Ve + Rbcde ∇a Ve = ∇a Rbcde Ve + Rae Rbcde + 1 Rbcda .

This proves the lemma. 1. 1. Since the solution σ(x, t) of the (PDE) is uniformly bounded with respect to the bundle metric hab on M × [t0 , T ] by hypothesis, we may assume that K is contained in a tubular neighborhood V (r) of the zero section in V whose intersection with each fiber Vx is a ball of radius r around the origin measured by the bundle metric hab for some large r > 0. Recall that gij (·, t), t ∈ [0, T ], is a smooth solution to the Ricci flow with uniformly bounded curvature on M × [0, T ].

M This finishes the proof of the proposition. 6) λ(gij ) = inf F (gij , f ) | f ∈ C ∞ (M ), e−f dV = 1 . M If we set u = e−f /2 , then the functional F can be expressed in terms of u as F= (Ru2 + 4|∇u|2 )dV, M THE HAMILTON-PERELMAN THEORY OF RICCI FLOW 203 and the constraint M e−f dV = 1 becomes M u2 dV = 1. Therefore λ(gij ) is just the first eigenvalue of the operator −4∆ + R. Let u0 > 0 be a first eigenfunction of the operator −4∆ + R satisfying −4∆u0 + Ru0 = λ(gij )u0 . The f0 = −2 log u0 is a minimizer: λ(gij ) = F (gij , f0 ).

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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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