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

By Huai-Dong Cao, Xi-Ping Zhu.

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Extra 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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One can further show that u is positive (see [108]). Then the standard regularity theory of elliptic PDEs shows that u is smooth. We refer the reader to Rothaus [108] for more details. 13) τ (2∆f − |∇f |2 + R) + f − n = µ(gij , τ ). 9. (i) µ(gij (t), τ −t) is nondecreasing along the Ricci flow; moveover, the monotonicity is strict unless we are on a shrinking gradient soliton; (ii) A shrinking breather is necessarily a shrinking gradient soliton. Proof. Fix any time t0 , let f0 be a minimizer of µ(gij (t0 ), τ − t0 ).

Then ϕ+N (ϕ, t) ∈ Tϕ Z if and only if l(N (ϕ, t)) ≤ 0 for all l ∈ Sϕ Z. Suppose l(N (ϕ, t)) > 0 for some ϕ ∈ ∂Z and some l ∈ Sϕ Z. 1) cannot remain in Z. To see the converse, first note that we may assume Z is compact. This is because we can modify the vector field N (ϕ, t) by multiplying a cutoff function which is everywhere nonnegative, equals one on a large ball and equals zero on the complement of a larger ball. The paths of solutions of the ODE are unchanged inside the first large ball, so we can intersect Z with the second ball to make Z convex and compact.

By the standard strong maximum principle, we know that ρ(x, t) is positive everywhere in Ω for all t ∈ (0, T ]. For every ε > 0, we claim that at every point (x, t) ∈ Ω × [0, T ], there holds n−l+1 Mαβ (x, t)viα viβ + εet > ρ(x, t) i=1 for any (n − l + 1) orthogonal unit vectors {v1 , . . , vn−l+1 } at x. We argue by contradiction. Suppose not, then for some ε > 0, there will be a first time t0 > 0 and some (n − l + 1) orthogonal unit vectors {v1 , . . , vn−l+1 } at some point x0 ∈ Ω so that n−l+1 Mαβ (x0 , t0 )viα viβ + εet0 = ρ(x0 , t0 ) i=1 Let us extend each vi (i = 1, .