Basic geometry by George D. Birkhoff

By George D. Birkhoff

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1 applies. Now, having checked the claim, we can prove the theorem by applying the claim 2 of the previous theorem to the appropriate segment of the shortest geodesic, connecting x and x0 . 3 Theorem. For any w > 0 there exist τ = τ (w) > 0, K = K(w) < ∞, ρ = ρ(w) > 0 with the following property. Suppose we have a solution gij (t) to the Ricci flow, defined on M × [0, T ), where M is a closed threemanifold, and a point (x0 , t0 ), such that the ball B(x0 , r0 ) at t = t0 has volume ≥ wr0n , and sectional curvatures ≥ −r0−2 at each point.

N. Assume first that for each point in B(p, T0 , A0 h), where A0 = ǫ(δ ′ )−1 , the solution is defined on [t0 , t1 ]. 3) and the choice of △t we have a uniform curvature bound on this set for h−2 -scaled metric. Therefore we can define A1 , depending only on A0 and tending to infinity with A0 , such that the solution in P (p, T0 , A1 h, t1 − t0 ) is, after scaling and time shifting, A−1 1 close to the corresponding subset in the standard solution. In particular, the 9 scalar curvature on this subset does not exceed 2Qh−2 .

We can imagine two possibilities for the behavior of the curvature along γ in the scaled metric: either it stays bounded at bounded distances from x¯, or not. In the first case we can take a limit (for a subsequence) of the scaled metrics along γ and get a nonnegatively curved almost cylindrical metric, with γ going to infinity. Clearly, in this case the curvature at any point of the limit does not exceed c−1 ; therefore, the point z must have escaped to infinity, and the conclusion of the claim stands.

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