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By A Grothendieck

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Justification of the canonical neighborhood assumption requires additional arguments. In fact, we are able to construct solutions satisfying this assumption only allowing r and δ be functions of time rather than constants; clearly, the arguments of the previous section are valid in this case, if we assume that r(t), δ(t) are non-increasing, and bounded away from zero on every finite time interval. Proposition. ( Recall that we have excluded from the list of canonical neighborhoods the closed manifolds, ǫ-close to metric quotients of the round sphere.

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.

A ball B(x, t, ǫ−1 r) is called an ǫ-neck, if, after scaling the metric with factor −2 r , it is ǫ-close to the standard neck S2 × I, with the product metric, where S2 has constant scalar curvature one, and I has length 2ǫ−1 ; here ǫ-close refers to C N topology, with N > ǫ−1 . Petersburg 191011, Russia. edu 1 time t′ ∈ [−1, 0] has length 2ǫ−1 and scalar curvature (1 − t′ )−1 . A metric on S2 × I, such that each point is contained in some ǫ-neck, is called an ǫ-tube, or an ǫ-horn, or a double ǫ-horn, if the scalar curvature stays bounded on both ends, stays bounded on one end and tends to infinity on the other, and tends to infinity on both ends, respectively.

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A general theory of fibre spaces with structure sheaf (National Science Foundation research project on geometry of function space : report) by A Grothendieck


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