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An extension of Perelman’s soul theorem for singular space
, 2010
"... We will use the first and second variational formulae of lengthfunctional to establish an extension of Perelman’s soul theorem for singular spaces: “Let X be a complete, noncompact, finite dimensional Alexandrov space of nonnegative curvature. Suppose that X has no boundary and has positive curv ..."
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We will use the first and second variational formulae of lengthfunctional to establish an extension of Perelman’s soul theorem for singular spaces: “Let X be a complete, noncompact, finite dimensional Alexandrov space of nonnegative curvature. Suppose that X has no boundary and has positive curvature on a nonempty open subset. Then it must be contractible”.
Collapsing threedimensional closed Alexandrov spaces with a lower curvature bound
, 2012
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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.
LONGTIME BEHAVIOR OF 3 DIMENSIONAL RICCI FLOW D: PROOF OF THE MAIN RESULTS
"... Abstract. This is the fourth and last part of a series of papers on the longtime behavior of 3 dimensional Ricci flows with surgery. In this paper, we prove our main two results. The first result states that if the surgeries are performed correctly, then the flow becomes nonsingular eventually and ..."
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Abstract. This is the fourth and last part of a series of papers on the longtime behavior of 3 dimensional Ricci flows with surgery. In this paper, we prove our main two results. The first result states that if the surgeries are performed correctly, then the flow becomes nonsingular eventually and the curvature is bounded by Ct−1. The second result provides a qualitative description of the geometry as t→∞. Contents
SOUL THEOREM FOR 4DIMENSIONAL TOPOLOGICALLY REGULAR OPEN NONNEGATIVELY CURVED ALEXANDROV SPACES
"... Abstract. In this paper, we study the topology of topologically regular 4dimensional open nonnegatively curved Alexandrov spaces. These spaces occur naturally as the blowup limits of compact Riemannian manifolds with lower curvature bound. These manifolds have also been studied by Yamaguchi in h ..."
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Abstract. In this paper, we study the topology of topologically regular 4dimensional open nonnegatively curved Alexandrov spaces. These spaces occur naturally as the blowup limits of compact Riemannian manifolds with lower curvature bound. These manifolds have also been studied by Yamaguchi in his preprint [Yam02]. Our main tools are gradient flows of semiconcave functions and critical point theory for distance functions, which have been used to study the 3dimensional collapsing theory in the paper [CaoG10]. The results of this paper will be used in our future studies of collapsing 4manifolds, which will be discussed elsewhere. 0.