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K3 surfaces and string duality
"... The primary purpose of these lecture notes is to explore the moduli space of type IIA, type IIB, and heterotic string compactified on a K3 surface. The main tool which is invoked is that of string duality. K3 surfaces provide a fascinating arena for string compactification as they are not trivial sp ..."
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Cited by 62 (14 self)
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The primary purpose of these lecture notes is to explore the moduli space of type IIA, type IIB, and heterotic string compactified on a K3 surface. The main tool which is invoked is that of string duality. K3 surfaces provide a fascinating arena for string compactification as they are not trivial spaces but are sufficiently simple for one to be able to analyze most of their properties in detail. They also make an almost ubiquitous appearance in the common statements concerning string duality. We review the necessary facts concerning the classical geometry of K3 surfaces that will be needed and then we review “old string theory ” on K3 surfaces in terms of conformal field theory. The type IIA string, the type IIB string, the E8 × E8 heterotic string, and Spin(32)/Z2 heterotic string on a K3 surface are then each analyzed in turn. The discussion is biased in favour of purely geometric notions concerning the K3 surface
PseudoRiemannian metrics with parallel spinor fields and vanishing Ricci tensor
 In Global analysis and harmonic analysis (2000
"... by ..."
An equation of MongeAmpère type in conformal geometry, and fourmanifolds of positive Ricci curvature
, 2004
"... ..."
FOURMANIFOLDS WITHOUT EINSTEIN METRICS
 MATHEMATICAL RESEARCH LETTERS 3, 133–147 (1996)
, 1996
"... It is shown that there are infinitely many compact simply connected smooth 4manifolds which do not admit Einstein metrics, but nevertheless satisfy the strict HitchinThorpe inequality 2χ>3τ. The examples in question arise as nonminimal complex algebraic surfaces of general type, and the method ..."
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Cited by 53 (13 self)
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It is shown that there are infinitely many compact simply connected smooth 4manifolds which do not admit Einstein metrics, but nevertheless satisfy the strict HitchinThorpe inequality 2χ>3τ. The examples in question arise as nonminimal complex algebraic surfaces of general type, and the method ofproofstems from SeibergWitten theory.
On Nearly Parallel G_2Structures
, 1995
"... A nearly parallel G 2 structure on a 7dimensional Riemannian manifold is equivalent to a spin structure with a Killing spinor. We prove general results about the automorphism group of such structures and we construct new examples. We classify all nearly parallel G 2 manifolds with large symmetry ..."
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Cited by 46 (5 self)
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A nearly parallel G 2 structure on a 7dimensional Riemannian manifold is equivalent to a spin structure with a Killing spinor. We prove general results about the automorphism group of such structures and we construct new examples. We classify all nearly parallel G 2 manifolds with large symmetry group and in particular all homogeneous nearly parallel G 2 structures.
Fredholm operators and Einstein metrics on conformally compact manifolds
"... Abstract. The main result of this paper is the existence of asymptotically hyperbolic Einstein metrics with prescribed conformal infinity sufficiently close to that of a given asymptotically hyperbolic Einstein metric with nonpositive curvature. If the conformal infinities are sufficiently smooth, t ..."
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Cited by 42 (2 self)
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Abstract. The main result of this paper is the existence of asymptotically hyperbolic Einstein metrics with prescribed conformal infinity sufficiently close to that of a given asymptotically hyperbolic Einstein metric with nonpositive curvature. If the conformal infinities are sufficiently smooth, the resulting Einstein metrics have optimal Hölder regularity at the boundary. The proof is based on sharp Fredholm theorems for selfadjoint geometric linear elliptic operators on asymptotically hyperbolic manifolds. 1.
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.
The geometry of dynamical triangulations
 Lecture Notes in Physics m50
, 1997
"... The express purpose of these Lecture Notes is to go through some aspects of the simplicial quantum gravity model known as the Dynamical Triangulations approach. Emphasis has been on lying the foundations of the theory and on illustrating its subtle and often unexplored connections with many distinct ..."
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Cited by 35 (3 self)
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The express purpose of these Lecture Notes is to go through some aspects of the simplicial quantum gravity model known as the Dynamical Triangulations approach. Emphasis has been on lying the foundations of the theory and on illustrating its subtle and often unexplored connections with many distinct mathematical fields ranging from global riemannian geometry, moduli theory, number theory, and topology. Our exposition will concentrate on these points so that graduate students may find in these notes a useful exposition of some of the rigorous results one can establish in this field and hopefully a source of inspiration for new exciting problems. We also illustrate the deep and beautiful interplay between the analytical aspects of dynamical triangulations and the results of MonteCarlo simulations. The techniques described here are rather novel and allow us to address successfully many high points of great current interest in the subject of simplicial quantum gravity while requiring very lit1 tle in the way of fancy field theoretical arguments. As a consequence, these
L 2 curvature and volume renormalization of the AHE metrics on 4manifolds
 Math. Res. Lett
"... Abstract. This paper relates the boundary term in the ChernGaussBonnet formula on 4manifolds M with the renormalized volume V, as defined in the AdS/CFT correspondence, for asymptotically hyperbolic Einstein metrics on M. In addition we compute and discuss the differential or variation dV of V, o ..."
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Cited by 34 (6 self)
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Abstract. This paper relates the boundary term in the ChernGaussBonnet formula on 4manifolds M with the renormalized volume V, as defined in the AdS/CFT correspondence, for asymptotically hyperbolic Einstein metrics on M. In addition we compute and discuss the differential or variation dV of V, or equivalently the variation of the L 2 norm of the Weyl curvature, on the space of such Einstein metrics. 0. Introduction. The ChernGaussBonnet formula for a compact Riemannian 4manifold (M,g) without boundary states that