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A survey of the spherical space form problem by J. F. Davis

By J. F. Davis

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In this case there exists D > 0, such that for any y satisfying d = distt0 (x0 , y) > D, one can find x satisfying distt0 (x, y) = d, distt0 (x, x0 ) > 32 d. We claim that the scalar curvature R(y, t) is uniformly bounded for all such y and all t ∈ (t′ , t0 ]. Indeed, if R(y, t) is large, then the neighborhood of (y, t) is like in an ancient solution; therefore, (long) shortest geodesics γ and γ0 , connecting at time t the point y to x and x0 respectively, make the angle close to 0 or π at y; the former case is ruled out by the assumptions on distances, if D > 10C; in the latter case, x and x0 are separated at time t by a small neighborhood of y, 1 with diameter of order R(y, t)− 2 , hence the same must be true at time t0 , which is impossible if R(y, t) is too large.

Since x lies deeply inside an ǫ-horn, its canonical neighborhood is a strong ǫ-neck. Now Claim 2 gives the curvature estimate that allows us to take a limit α of appropriate scalings of the metrics gij on [T − h2 (xα ), T ] about xα , for a subsequence of α → ∞. By shifting the time parameter we may assume that the limit is defined on [−1, 0]. Clearly, for each time in this interval, the limit is a complete manifold with nonnegative sectional curvature; moreover, since xα was α contained in an ǫ-horn with boundary in Ωα ρ , and h(x )/ρ → 0, this manifold has two ends.

For any Q < ∞ there exists θ = θ(Q), 0 < θ < 1 with the following property. Suppose we are in the situation of the lemma above, ¯ with δ < δ(A, θ), A > ǫ−1 . Suppose that for some point x ∈ B(p, T0 , Ah) the solution is defined at x (at least) on [T0 , Tx ], Tx ≤ T, and satisfies Q−1 R(x, t) ≤ R(x, Tx ) ≤ Q(Tx − T0 )−1 for all t ∈ [T0 , Tx ]. Then Tx ≤ T0 + θh2 . Proof. Indeed, if Tx > T0 + θh2 , then by lemma R(x, T0 + θh2 ) ≥ const · (1 − θ)−1 h−2 , whence R(x, Tx ) ≥ const · Q−1 (1 − θ)−1 h−2 , and Tx − T0 ≤ const · Q2 (1 − θ)h2 < θh2 if θ is close enough to one.

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