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How Riemannian manifolds converge
 PROCEEDINGS OF THE INTERNATIONAL CONFERENCE ON METRIC AND DIFFERENTIAL GEOMETRY IN TIANJING AND BEIJING
, 2010
"... This is an intuitive survey of extrinsic and intrinsic notions of convergence of manifolds complete with pictures of key examples and a discussion of the properties associated with each notion. We begin with a description of three extrinsic notions which have been applied to study sequences of subm ..."
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This is an intuitive survey of extrinsic and intrinsic notions of convergence of manifolds complete with pictures of key examples and a discussion of the properties associated with each notion. We begin with a description of three extrinsic notions which have been applied to study sequences of submanifolds in Euclidean space: Hausdorff convergence of sets, flat convergence of integral currents, and weak convergence of varifolds. We next describe a variety of intrinsic notions of convergence which have been applied to study sequences of compact Riemannian manifolds: GromovHausdorff convergence of metric spaces, convergence of metric measure spaces, Instrinsic Flat convergence of integral current spaces, and ultralimits of metric spaces. We close with a speculative section addressing possible notions of intrinsic varifold convergence, convergence of Lorentzian manifolds and area convergence.
How Riemannian Manifolds Converge: a Survey
 Progress in Mathematics
, 2010
"... ABSTRACT: This is an intuitive survey of extrinsic and intrinsic notions of convergence of manifolds complete with pictures of key examples and a discussion of the properties associated with each notion. We begin with a description of three extrinsic notions which have been applied to study sequence ..."
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ABSTRACT: This is an intuitive survey of extrinsic and intrinsic notions of convergence of manifolds complete with pictures of key examples and a discussion of the properties associated with each notion. We begin with a description of three extrinsic notions which have been applied to study sequences of submanifolds in Euclidean space: Hausdorff convergence of sets, flat convergence of integral currents, and weak convergence of varifolds. We next describe a variety of intrinsic notions of convergence which have been applied to study sequences of compact Riemannian manifolds: GromovHausdorff convergence of metric spaces, convergence of metric measure spaces, Instrinsic Flat convergence of integral current spaces, and ultralimits of metric spaces. We close with a speculative section addressing possible notions of intrinsic varifold convergence, convergence of Lorentzian manifolds and area convergence. 1.
DEGENERATIONS OF RIEMANNIAN MANIFOLDS
, 2007
"... 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/noncollapsed regions in Riemannian manifold under various curvature assumptions. Numerous application ..."
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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/noncollapsed 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], Colding [Col98],