Results 1  10
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188
Gauge theory for embedded surfaces
 I, Topology
, 1993
"... (i) Topology of embedded surfaces. Let X be a smooth, simplyconnected 4manifold, and ξ a 2dimensional homology class in X. One of the features of topology in dimension 4 is the fact that, although one may always represent ξ as the fundamental class of some smoothly ..."
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Cited by 68 (6 self)
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(i) Topology of embedded surfaces. Let X be a smooth, simplyconnected 4manifold, and ξ a 2dimensional homology class in X. One of the features of topology in dimension 4 is the fact that, although one may always represent ξ as the fundamental class of some smoothly
An equation of MongeAmpère type in conformal geometry, and fourmanifolds of positive Ricci curvature
, 2004
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Towards the Poincaré Conjecture and the Classification of 3Manifolds
, 2003
"... The Poincaré Conjecture was posed ninetynine years ago and may possibly have been proved in the last few months. This note will be an account of some of the major results over the past hundred years which have paved the way towards a proof and towards the even more ambitious project of classifying a ..."
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Cited by 27 (0 self)
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The Poincaré Conjecture was posed ninetynine years ago and may possibly have been proved in the last few months. This note will be an account of some of the major results over the past hundred years which have paved the way towards a proof and towards the even more ambitious project of classifying all compact 3dimensional manifolds. The final paragraph provides a brief description of the latest developments, due to Grigory Perelman. A more serious discussion of Perelman’s work will be provided in a subsequent note by Michael Anderson.
On the moduli space of diffeomorphic algebraic surfaces
 Invent. Math
"... Abstract. In this paper we show that the number of deformation types of complex structures on a fixed smooth oriented fourmanifold can be arbitrarily large. The examples that we consider in this paper are locally simple abelian covers of rational surfaces. The proof involves the algebraic descripti ..."
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Cited by 25 (2 self)
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Abstract. In this paper we show that the number of deformation types of complex structures on a fixed smooth oriented fourmanifold can be arbitrarily large. The examples that we consider in this paper are locally simple abelian covers of rational surfaces. The proof involves the algebraic description of rational blow down, classical BrillNoether theory and deformation theory of normal flat abelian covers. One of the main problems concerning the differential topology of algebraic surfaces leaving unsolved by the “SeibergWitten revolution ” was to determine whether the differential type of a compact complex surface determines the deformation type. Two compact complex manifolds have the same deformation type if they are fibres of a proper smooth family over a connected
Minimal entropy and collapsing with curvature bounded from below
 Invent. Math
"... Abstract. We show that if a closed manifold M admits an Fstructure (not necessarily polarized, possibly of rank zero) then its minimal entropy vanishes. In particular, this is the case if M admits a nontrivial S 1action. As a corollary we obtain that the simplicial volume of a manifold admitting ..."
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Cited by 25 (4 self)
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Abstract. We show that if a closed manifold M admits an Fstructure (not necessarily polarized, possibly of rank zero) then its minimal entropy vanishes. In particular, this is the case if M admits a nontrivial S 1action. As a corollary we obtain that the simplicial volume of a manifold admitting an Fstructure is zero. We also show that if M admits an Fstructure then it collapses with curvature bounded from below. This in turn implies that M collapses with bounded scalar curvature or, equivalently, its Yamabe invariant is nonnegative. We show that Fstructures of rank zero appear rather frequently: every compact complex elliptic surface admits one as well as any simply connected closed 5manifold. We use these results to study the minimal entropy problem. We show the following two theorems: suppose that M is a closed manifold obtained by taking connected sums of copies of S 4, CP 2, CP 2, S 2 × S 2 and the K3 surface. Then M has zero minimal entropy. Moreover, M admits a smooth Riemannian metric with zero topological entropy if and only if M is diffeomorphic to S 4, CP 2, S 2 × S 2, CP 2 #CP 2 or CP 2 #CP 2. Finally, suppose that M is a closed simply connected 5manifold. Then M has zero minimal entropy. Moreover, M admits a smooth Riemannian metric with zero topological entropy if and only if M is diffeomorphic to S 5, S 3 ×S 2, the nontrivial S 3bundle over S 2 or the Wumanifold SU(3)/SO(3). 1.
Real root conjecture fails for five and higherdimensional spheres, Discrete Comput
 Geom
"... Abstract: A construction of convex flag triangulations of five and higher dimensional spheres, whose hpolynomials fail to have only real roots, is given. We show that there is no such example in dimensions lower than five. A condition weaker than realrootedness is conjectured and some evidence is p ..."
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Cited by 24 (1 self)
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Abstract: A construction of convex flag triangulations of five and higher dimensional spheres, whose hpolynomials fail to have only real roots, is given. We show that there is no such example in dimensions lower than five. A condition weaker than realrootedness is conjectured and some evidence is provided. Let the fpolynomial fX of a simplicial complex X be defined by the formula fX(t): = ∑ t #σ. There is a classical problem: what can be said in general about the fpolynomials of (a certain class of) simplicial complexes? In particular, it is well known what polynomials appear as fpolynomials of • general simplicial complexes, or • triangulations of spheres that are the boundary complexes of convex polytopes (the reader may consult [St1] for ample discussion). The question concerning all triangulations of spheres remains still open. However the answer is conjecturally the same. What we are interested in is the special case of the latter. Namely, what can be said in general about
Asymptotic stability equals exponential stability, and ISS equals finite energy gain  if you twist your eyes
, 1999
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Geometric topology: localization, periodicity and Galois symmetry
 KMonographs in Math. 8
, 2005
"... (The 1970 MIT notes) ..."
A Uniformization Theorem Of Complete Noncompact Kähler Surfaces With Positive Bisectional Curvature
, 2002
"... In this paper, by combining techniques from Ricci flow and algebraic geometry, we prove the following generalization of the classical uniformization theorem of Riemann surfaces. Given a complete noncompact complex two dimensional Kähler manifold M of positive and bounded holomorphic bisectional curv ..."
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Cited by 18 (5 self)
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In this paper, by combining techniques from Ricci flow and algebraic geometry, we prove the following generalization of the classical uniformization theorem of Riemann surfaces. Given a complete noncompact complex two dimensional Kähler manifold M of positive and bounded holomorphic bisectional curvature, suppose its geodesic balls have Euclidean volume growth and its scalar curvature decays to zero at infinity in the average sense, then M is biholomorphic to C². During the proof, we also discover an interesting gap phenomenon which says that a Kähler manifold as above automatically has quadratic curvature decay at infinity in the average sense.
On complete noncompact Kähler manifolds with positive bisectional curvature, preprint
, 2000
"... We prove that a complete noncompact Kähler manifold M n of positive bisectional curvature satisfying suitable growth conditions is biholomorphic to a pseudoconvex domain of C n and we show that the manifold is topologically R 2n. In particular, when M n is a Kähler surface of positive bisectional cu ..."
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Cited by 18 (3 self)
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We prove that a complete noncompact Kähler manifold M n of positive bisectional curvature satisfying suitable growth conditions is biholomorphic to a pseudoconvex domain of C n and we show that the manifold is topologically R 2n. In particular, when M n is a Kähler surface of positive bisectional curvature satisfying certain natural geometric growth conditions, it is biholomorphic to C 2. 1.