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Noncollapsed Gromov-Hausdorff Limit Spaces with Ricci Curvature Bounded Below
By a fundamental (relatively soft) theorem of Gromov, the collection of riemannian manifolds (Mn, g) with Ricci curvature bounded below, say RicMn ≥ −(n − 1), and diameter ≤ d, is precompact in the Gromov-Hausdorff topology. Thus, any sequence of such manifolds has a subsequence (Mni, gi) which converges in a weak geometric sense to some limiting metric space (X, d). Intuitively, this means that no matter how acute our vision, if i is sufficiently large, we will be unable to distinguish between (Mni, gi) and (X, d).
The limits of such sequences can be thought of as counter parts of distributions or Sobolev functions. Thus, even for applications to the smooth case, it is important to understand their structure. In the noncollapsing case, where by definition, Xn has Hausdorff dimension n, the structure is much more constrained. From our work with Colding in the mid 1990’s, it is known that for all > 0, there is a closed subset S ⊂ Xn of Hausdorff dimension ≤ n−2, such that Xn\S is θ( )-bi-Holder equivalent to a smooth ¨ n-dimensional riemannian manifold. Moreover, θ( ) → 1 as → 0. Until relatively recently (2018) little else was known about the structure of S .
We will discuss some structural results obtained in joint work with Aaron Naber an Wenshuai Jiang. They show that in several precise senses, S strongly resembles a smooth submanifold of dimension ≤ n − 2. In particular, if S has postive (n − 2)-dimensional Hausdorff measure, then it is (n − 2)-rectifiable, which is the measure theoretic version of being a manifold of dimension n − 2. Examples of Naber and Nan Li show that in actuality, S need have no manifold points.