Geometric Structures of Collapsing Riemannian Manifolds I

Abstract

Let (M^n_i,g_i,p_i) be a sequence of smooth pointed complete n-dimensional Riemannian Manifolds with uniform bounds on the sectional curvatures and let (X,d,p) be a metric space such that (M^n_i,g_i,p_i) -> (X,d,p) in the Gromov-Hausdorff sense. Let O \subseteq X be the set of points x \in X such that there exists a neighborhood of x which is isometric to an open set in a Riemannian orbifold and let B = O^c be the complement set. Then we have the sharp estimates dim_Haus(B) \leq min{n-5, dim_Haus(X)-3}, and further for arbitrary x \in X we have that x \in O iff a neighborhood of x has bounded Alexandroff curvature. In particular, if n \leq 4 then B is empty and (X,d) is a Riemannian orbifold. Our main application is to prove that a collapsed limit of Einstein four manifolds has a smooth Riemannian orbifold structure away from a finite number of points, and that near these points the curvatures has a -dist^{-2} lower bound.

35 Pages

Cited by in corpus (9)

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• Geometric Structures of Collapsing Riemannian Manifolds II: N*-bundles and Almost Ricci Flat Spaces
• Perelman's collapsing theorem for 3-manifolds
• Harnack Inequalities for Critical 4-manifolds with a Ricci Curvature Bound
• Curvature Estimates for Critical 4-Manifolds with a Lower Ricci Curvature Bound
• Regularity and convergence of 4-dimensional extremal Kahler metrics