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Universal bounds for hyperbolic Dehn surgery
 Annals of Math
, 2005
"... Abstract. This paper gives a quantitative version of Thurston’s hyperbolic Dehn surgery theorem. Applications include the first universal bounds on the number of nonhyperbolic Dehn fillings on a cusped hyperbolic 3manifold, and estimates on the changes in volume and core geodesic length during hyp ..."
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Cited by 40 (2 self)
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Abstract. This paper gives a quantitative version of Thurston’s hyperbolic Dehn surgery theorem. Applications include the first universal bounds on the number of nonhyperbolic Dehn fillings on a cusped hyperbolic 3manifold, and estimates on the changes in volume and core geodesic length during hyperbolic Dehn filling. The proofs involve the construction of a family of hyperbolic conemanifold structures, using infinitesimal harmonic deformations and analysis of geometric limits. 1.
Small curvature surfaces in hyperbolic 3manifolds
 JOURNAL OF KNOT THEORY AND ITS RAMIFICATIONS
"... In a paper of Menasco and Reid, it is conjectured that there exist no hyperbolic knots in S³ for which the complement contains a closed embedded totally geodesic surface. In this note, we show that one can get ”as close as possible” to a counterexample. Specifically, we construct a sequence of hype ..."
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Cited by 9 (0 self)
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In a paper of Menasco and Reid, it is conjectured that there exist no hyperbolic knots in S³ for which the complement contains a closed embedded totally geodesic surface. In this note, we show that one can get ”as close as possible” to a counterexample. Specifically, we construct a sequence of hyperbolic knots {Kn} with complements containing closed embedded essential surfaces having principal curvatures converging to zero as n tends to infinity. We also construct a family of twocomponent links for which the complements contain closed embedded totally geodesic surfaces of arbitrarily large genera. In addition, we prove that a closed embedded surface with sufficiently small principal curvatures is not only quasiFuchsian (a result of W. Thurston’s), but it is also either acylindrical or or the boundary of a twisted Ibundle.
Dynamics of the mapping class group action on the variety of PSL(2,C) characters
 math.GT/0504474 (submitted). GROUP REPRESENTATIONS 33
"... Abstract. We study the action of the mapping class group Mod(S) on the boundary ∂Q of quasifuchsian space Q. Among other results, Mod(S) is shown to be topologically transitive on the subset C ⊂ ∂Q of manifolds without a conformally compact end. We also prove that any open subset of the character va ..."
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Cited by 9 (2 self)
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Abstract. We study the action of the mapping class group Mod(S) on the boundary ∂Q of quasifuchsian space Q. Among other results, Mod(S) is shown to be topologically transitive on the subset C ⊂ ∂Q of manifolds without a conformally compact end. We also prove that any open subset of the character variety X(π1(S), PSL2 C) intersecting ∂Q does not admit a nonconstant Mod(S)invariant meromorphic function. This is related to a question of Goldman. 1.
Completion of the proof of the geometrization conjecture
"... This paper builds upon and is an extension of [13]. In this paper, we complete a proof of the following: Geometrization Conjecture: Any closed, orientable, prime 3manifold M contains a disjoint union of embedded 2tori and Klein bottles such that each connected component of the complement admits a ..."
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Cited by 6 (0 self)
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This paper builds upon and is an extension of [13]. In this paper, we complete a proof of the following: Geometrization Conjecture: Any closed, orientable, prime 3manifold M contains a disjoint union of embedded 2tori and Klein bottles such that each connected component of the complement admits a locally homogeneous Riemannian metric of finite volume. Recall that a Riemannian manifold is homogeneous if its isometry group acts transitively on the underlying manifold; a locally homogeneous Riemannian manifold is the quotient of a homogeneous Riemannian manifold by a discrete group of isometries acting freely. Recall also that a prime 3manifold is one which is not diffeomorphic to S 3 and which is not a connected sum of two manifolds neither of which is diffeomorphic to S 3. It is a classic result in 3manifold topology, see [12] that every 3manifold is a connected sum of a finite number of prime 3manifolds, and this decomposition is unique up to the order of the factors. The main part of this paper is devoted to giving a proof of Theorem 7.4 stated
The corank conjecture for 3manifold groups
 Algebr. Geom. Topol
"... Abstract In this paper we construct explicit examples of both closed and noncompact finite volume hyperbolic manifolds which provide counterexamples to the conjecture that the corank of a 3manifold group (also known as the cut number) is bounded below by onethird the first Betti number. AMS Clas ..."
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Abstract In this paper we construct explicit examples of both closed and noncompact finite volume hyperbolic manifolds which provide counterexamples to the conjecture that the corank of a 3manifold group (also known as the cut number) is bounded below by onethird the first Betti number. AMS Classification 57M05; 57M50, 20F34
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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Cited by 3 (0 self)
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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
THE NATURE OF CONTEMPORARY CORE MATHEMATICS
, 2010
"... Abstract. The goal of this essay is a description of modern mathematical practice, with emphasis on differences between this and practices in the nineteenth century. I explain how and why these differences greatly increased the effectiveness of mathematical methods and enabled sweeping developments ..."
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Abstract. The goal of this essay is a description of modern mathematical practice, with emphasis on differences between this and practices in the nineteenth century. I explain how and why these differences greatly increased the effectiveness of mathematical methods and enabled sweeping developments in the twentieth century. A particular concern is the significance for mathematics education: elementary education remains modeled on the mathematics of the nineteenth century and before, and use of modern methodologies might give advantages similar to those seen in mathematics. This draft is about 90 % complete, and comments are welcome. 1.
ATG The corank conjecture for 3manifold groups
, 2002
"... Abstract In this paper we construct explicit examples of both closed and noncompact finite volume hyperbolic manifolds which provide counterexamples to the conjecture that the corank of a 3manifold group (also known as the cut number) is bounded below by onethird the first Betti number. AMS Clas ..."
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Abstract In this paper we construct explicit examples of both closed and noncompact finite volume hyperbolic manifolds which provide counterexamples to the conjecture that the corank of a 3manifold group (also known as the cut number) is bounded below by onethird the first Betti number. AMS Classification 57M05; 57M50, 20F34