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Thus, the formation of singularities in the limit reflects the absence of uniform control over the scale on which the metric becomes standard. sdorff dimension of the singular set of Y. From this it follows that uniform control of the metric does exist, except perhaps on a set which is very small in a definite sense. re of the singular set. These imply that there is a weaker sort of uniform control at all points of all manifolds in the approximating sequence. Imagine that a manifold with RicM" ~ -(n-l) and diam(Mn) = d, is observed under a powerful microscope, so that all distances appear to have been multiplied by a large factor c 1 • A ball of radius £ will appear to have radius 1 and the Ricci curvature will appear to be bounded below by - (n - 1) .

In the Einstein case, we can assume that fen) has been chosen such that for f :5 fen), we have R. = R. 2. \ 0 N oncollapsed tangent cones are metric cones. 4. 3). Then every (possibly iterated) tangent cone is a metric cone, C(Z), on some length space, X, with diam(X) :5 71". Blow up arguments. 3). ' ~ -(n-l), to be proved by "blow up" arguments; [Fe]. These are proofs by contr;Wiction, in which the main step consists of showing that if a desired property were ever to fail, it would already fail for some tangent cone.

4) which are metric cones, the above complications do not occur. Roughly speaking, an almost parallel vector field on a cone is always almost the gradient of the coordinate function corresponding to the factor, R, of some (global) almost isometric splitting. 32 of [Ch2j. 9. 3) and assume that is Kahler for all i. Then S2H1 = S2i for all i. In particular, all strata have even codimension. Moreover, every tangent cone, Y II , has a parallel almost complex structure. If YII = Ri x C(X) denotes the isometric splitting for which j is maximal, then Ri = C!.