Degenerations of Riemannian Manifolds
نویسنده
چکیده
This is an expositiry article on collapsing theory written for the Modern Encyclopedia of Mathematical Physics (MEMPhys). We focus on describing the geometric and topological structure of collapsed/non-collapsed regions in Riemannian manifold under various curvature assumptions. Numerous applications of collapsing theory to Riemannian geometry are not discussed in this survey, due to page limits dictated by the encyclopedia format. More information on collapsing can be found in the ICM articles by Perelman [Per95], Petrunin [Pet02], Rong [Ron02], in Cheeger’s book on Cheeger-Colding theory [Che01], and in the comprehensive survey of Fukaya [Fuk06]. The article ends with an appendix on Gromov-Hausdorff distance, also written for MEMPhys. A fundamental problem in Riemannian geometry is to analyze how a family of Riemannian manifolds can degenerate. For example, in the case of Einstein manifolds it is natural to study how the Einstein equation develops a singularity, or how to compactify the moduli space of Einstein metrics. The concept of a Gromov-Hausdorff convergence provides a general framework for studying metric degenerations. Let (Mk, pk) is a sequence of complete pointed n-dimensional Riemannian manifolds that Gromov-Hausdorff converge to the space (Y, q); in other words (Mk, pk) degenerates to (Y, q). Due to Gromov’s compactness theorem, a simple way to ensure that (Mk, pk) has a GromovHausdorff converging subsequence is to assume that Ric(Mk) ≥ c for some c . The sequence Mk is said to collapse near the points pk if the volumes of the unit balls centered at pk tend to zero as k → ∞ . Otherwise, Mk is called non-collapsing near pk . An additional challenge is that collapsing and noncollapsing may occur at the same time on different parts of the manifold (even though if Ric(Mk) ≥ c , then the distance between collapsed and non-collapsed parts of Mk has to go to infinity as k → ∞ because of Bishop-Gromov’s volume comparison). It is mainstream of global Riemannian geometry to study collapsing and non-collapsing sequence of manifolds under various assumptions on curvature such as |sec| ≤ C , or sec ≥ 0 or Ric ≥ c or |Ric| ≤ c , or
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