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21
Weak collapsing and geometrisation of aspherical 3manifolds
, 2008
"... Let M be a closed, orientable, irreducible, nonsimply connected 3manifold. We prove that if M admits a sequence of Riemannian metrics whose sectional curvature is locally controlled and whose thick part becomes asymptotically hyperbolic and has a sufficiently small volume, then M is Seifert fibred ..."
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Cited by 16 (4 self)
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Let M be a closed, orientable, irreducible, nonsimply connected 3manifold. We prove that if M admits a sequence of Riemannian metrics whose sectional curvature is locally controlled and whose thick part becomes asymptotically hyperbolic and has a sufficiently small volume, then M is Seifert fibred or contains an incompressible torus. This result gives an alternative approach for the last step in Perelman’s proof of the Geometrisation Conjecture for aspherical 3manifolds.
DEVELOPMENTS AROUND POSITIVE SECTIONAL CURVATURE
, 902
"... Abstract. This is not in any way meant to be a complete survey on positive curvature. Rather it is a short essay on the fascinating changes in the landscape surrounding positive curvature. In particular, details and many results and references are not included, and things are not presented in chrono ..."
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Cited by 10 (1 self)
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Abstract. This is not in any way meant to be a complete survey on positive curvature. Rather it is a short essay on the fascinating changes in the landscape surrounding positive curvature. In particular, details and many results and references are not included, and things are not presented in chronological order. Spaces of positive curvature have always enjoyed a particular role in Riemannian geometry. Classically, this class of spaces form a natural and vast extension of spherical geometry, and in the last few decades their importance for the study of general manifolds with a lower curvature bound via Alexandrov geometry has become apparent. The importance of Alexandrov geometry to Riemannian geometry stems from the fact that there are several natural geometric operations that are closed in Alexandrov geometry but not in Riemanian geometry. These include taking Gromov Hausdorff limits, taking quotients by isometric group actions, and forming joins of positively curved spaces. In particular, limits (or quotients) of Riemanian manifolds with a lower (sectional) curvature bound are Alexandrov spaces, and only rarely Riemannian manifolds. Analyzing limits frequently involves blow ups leading to spaces with nonnegative curvature as, e.g., in Perelman’s work on the geometrization conjecture. Also the infinitesimal structure of an Alexandrov space is expressed via its “tangent spaces”, which are cones on positively curved spaces. Hence the collection of all compact positively curved spaces (up to scaling) agrees with the class of all possible socalled spaces of directions, in Alexandrov spaces. So spaces of positive curvature play the same role in Alexandrov geometry as round spheres do in Riemannian geometry. In addition to positively, and nonnegativey curved spaces, yet another class of spaces has emerged in the general context of convergence under a lower curvature bound, namely almost nonnegatively curved spaces. These are spaces allowing metrics with diameter say 1, and lower curvature bound arbitrarily close to 0. They are expected to play a role among spaces with a lower curvature bound, analogous to that almost flat spaces play for spaces with bounded curvature. In summary, the following classes of spaces play essential roles in the study of spaces with a lower curvature bound:
A SIMPLE PROOF OF PERELMAN’S COLLAPSING THEOREM FOR 3MANIFOLDS
"... Abstract. We will simplify earlier proofs of Perelman’s collapsing theorem for 3manifolds given by ShioyaYamaguchi [SY00][SY05] and MorganTian [MT08]. A version of Perelman’s collapsing theorem states: “Let {M 3 i} be a sequence of compact Riemannian 3manifolds with curvature bounded from below ..."
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Cited by 8 (2 self)
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Abstract. We will simplify earlier proofs of Perelman’s collapsing theorem for 3manifolds given by ShioyaYamaguchi [SY00][SY05] and MorganTian [MT08]. A version of Perelman’s collapsing theorem states: “Let {M 3 i} be a sequence of compact Riemannian 3manifolds with curvature bounded from below by (−1) and diam(M 3 i) ≥ c0> 0. Suppose that all unit metric balls in M 3 i have very small volume at most is closed or it vi → 0 as i → ∞ and suppose that either M 3 i has possibly convex incompressible toral boundary. Then M 3 i must be a graphmanifold for sufficiently large i”. This result can be viewed as an extension of implicit function theorem. Among other things, we use Perelman’s critical point theory (e.g., multiple conic singularity theory and his fibration theory) for Alexandrov spaces to construct the desired local Seifert fibration structure on collapsed 3manifolds. The verification of Perelman’s collapsing theorem is the last step of Perelman’s proof of Thurston’s Geometrization Conjecture on the classification of 3manifolds. Our proof of Perelman’s collapsing theorem is accessible to nonexperts and advanced graduate students. Contents
An Optimal Extension of Perelman’s Comparison Theorem for Quadrangles and its Applications
"... In this paper we discuss an extension of Perelman’s comparison for quadrangles. Among applications of this new comparison theorem, we study the equidistance evolution of hypersurfaces in Alexandrov spaces with nonnegative curvature. We show that, in certain cases, the equidistance evolution of hype ..."
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Cited by 7 (4 self)
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In this paper we discuss an extension of Perelman’s comparison for quadrangles. Among applications of this new comparison theorem, we study the equidistance evolution of hypersurfaces in Alexandrov spaces with nonnegative curvature. We show that, in certain cases, the equidistance evolution of hypersurfaces become totally convex relative to a bigger subdomain. An optimal extension of 2nd variational formula for geodesics by Petrunin will be derived for the case of nonnegative curvature. In addition, we also introduced the generalized second fundament forms for subsets in Alexandrov spaces. Using this new notion, we will propose an approach to study two open problems in Alexandrov geometry.
Topology of Riemannian submanifolds with prescribed boundary
, 2008
"... We prove that a smooth compact immersed submanifold of codimension 2 in R n, n ≥ 3, bounds at most finitely many topologically distinct compact nonnegatively curved hypersurfaces. Analogous results for complete fillings of arbitrary Riemannian submanifolds are obtained as well. On the other hand, w ..."
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Cited by 4 (4 self)
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We prove that a smooth compact immersed submanifold of codimension 2 in R n, n ≥ 3, bounds at most finitely many topologically distinct compact nonnegatively curved hypersurfaces. Analogous results for complete fillings of arbitrary Riemannian submanifolds are obtained as well. On the other hand, we show that these finiteness theorems may not hold if the codimenion is too high, or the prescribed boundary is not sufficiently regular.
SPECTRAL AND GEOMETRIC BOUNDS ON 2ORBIFOLD DIFFEOMORPHISM TYPE
, 811
"... Abstract. We show that a Laplace isospectral family of two dimensional Riemannian orbifolds, sharing a lower bound on sectional curvature, contains orbifolds of only a finite number of orbifold category diffeomorphism types. We also show that orbifolds of only finitely many orbifold diffeomorphism t ..."
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Cited by 3 (1 self)
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Abstract. We show that a Laplace isospectral family of two dimensional Riemannian orbifolds, sharing a lower bound on sectional curvature, contains orbifolds of only a finite number of orbifold category diffeomorphism types. We also show that orbifolds of only finitely many orbifold diffeomorphism types may arise in any collection of 2orbifolds satisfying lower bounds on sectional curvature and volume, and an upper bound on diameter. An argument converting spectral data to geometric bounds shows that the first result is a consequence of the second.
Relatively maximum volume rigidity in Alexandrov geometry
 Pacific J. of Mathematics
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