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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”.
GEOMETRIC ANALYSIS
, 2005
"... This was a talk I gave in the occasion of the seventieth anniversary of the Chinese Mathematical Society. I dedicated it in memory of my teacher S. S. Chern who passed away half a year ago. During my graduate study, I was rather free in picking research topics. I [538] worked on fundamental groups o ..."
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This was a talk I gave in the occasion of the seventieth anniversary of the Chinese Mathematical Society. I dedicated it in memory of my teacher S. S. Chern who passed away half a year ago. During my graduate study, I was rather free in picking research topics. I [538] worked on fundamental groups of manifolds with nonpositive curvature. But in the second year of my study, I started to look into differential equations on manifolds. While Chern did not express much opinions on this part of my research, he started to appreciate it a few years later. In fact, after Chern gave a course on Calabi’s works on affine geometry in 1972 in Berkeley, Cheng told me these inspiring lectures. By 1973, Cheng and I started to work on some problems mentioned in his lectures. We did not realize that great geometers Pogorelov, Calabi and Nirenberg were also working on them. We were excited that we solved some of the conjectures of Calabi on improper affine spheres. But soon we found out that Pogorelov [398] published it right before us by different arguments. Nevertheless our ideas are useful to handle other problems in
COLLAPSING MANIFOLDS OBTAINED BY KUMMERTYPE CONSTRUCTIONS
, 2005
"... Abstract. We construct Fstructures on a Bott manifold and on some other manifolds obtained by Kummertype constructions. We also prove that if M = E#X where E is a fiber bundle with structure group G and a fiber admitting a Ginvariant metric of nonnegative sectional curvature and X admits an Fst ..."
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Abstract. We construct Fstructures on a Bott manifold and on some other manifolds obtained by Kummertype constructions. We also prove that if M = E#X where E is a fiber bundle with structure group G and a fiber admitting a Ginvariant metric of nonnegative sectional curvature and X admits an Fstructure with one trivial covering, then one can construct on M a sequence of metrics with sectional curvature uniformly bounded from below and volume tending to zero (i.e. VolK(M) = 0). As a corollary we prove that all the elements in the Spin cobordism ring can be represented by manifolds M with VolK(M) = 0. 1.
Volumecollapsed threemanifolds with a lower curvature bound
, 2008
"... In this paper we determine the topology of threedimensional complete orientable Riemannian manifolds with a uniform lower bound of sectional curvature whose volume is sufficiently small. ..."
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In this paper we determine the topology of threedimensional complete orientable Riemannian manifolds with a uniform lower bound of sectional curvature whose volume is sufficiently small.
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.
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.