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Variational theory for the total scalar curvature functional for Riemannian metrics and related topics
 in Topics in Calculus of Variations (Montecatini
, 1987
"... The contents of this paper correspond roughly to that of the author's lecture series given at Montecatini in July 1987. This paper is divided into five sections. In the first we present he EinsteinHilbert variationM problem on the space of Riemannian metrics on a compact closed manifold M. We ..."
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Cited by 177 (2 self)
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The contents of this paper correspond roughly to that of the author's lecture series given at Montecatini in July 1987. This paper is divided into five sections. In the first we present he EinsteinHilbert variationM problem on the space of Riemannian metrics on a compact closed manifold M. We compute the first and secol~d variation and observe the distinction which arises between conformal directions and their orthogonal complements. We discuss variational characterizations of constant curvalure m trics, and give a proof of 0bata's uniqueness theorem. Much of the material in this section can be found in papers of Berger Ebin [3], FischerMarsden [8], N. Koiso [14], and also in the recent book by A. Besse [4] where the reader will find additional references. In §2 we give a general discussion of the Yamabe problem and its resolution. We also give a detailed analysis of the solutions of the Yamabe equation for the product conformal structure on SI(T) x S~1(1), a circle of radius T crossed with a sphere of radius one. These exhibit interesting variational fea,tures uch a.s symmetry breaking and the existence of solutions with high Morse index. Since the time of the summer school in Montecatini, the beautiful survey paper of J. Lee and T. Parker [15] has appeared. This gives a detailed discussion of the
An equation of MongeAmpère type in conformal geometry, and fourmanifolds of positive Ricci curvature
, 2004
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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 88 (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
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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 78 (8 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.
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 meth ..."
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Cited by 77 (14 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.
Continuity, curvature, and the general covariance of optimal transportation
, 2008
"... Let M and ¯ M be ndimensional manifolds equipped with suitable Borel probability measures ρ and ¯ρ. For subdomains M and ¯ M of Rn, Ma, Trudinger & Wang gave sufficient conditions on a transportation cost c ∈ C4 (M × ¯ M) to guarantee smoothness of the optimal map pushing ρ forward to ¯ρ; the ..."
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Cited by 76 (20 self)
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Let M and ¯ M be ndimensional manifolds equipped with suitable Borel probability measures ρ and ¯ρ. For subdomains M and ¯ M of Rn, Ma, Trudinger & Wang gave sufficient conditions on a transportation cost c ∈ C4 (M × ¯ M) to guarantee smoothness of the optimal map pushing ρ forward to ¯ρ; the necessity of these conditions was deduced by Loeper. The present manuscript shows the form of these conditions to be largely dictated by the covariance of the question; it expresses them via nonnegativity of the sectional curvature of certain nullplanes in a novel but natural pseudoRiemannian geometry which the cost c induces on the product space M × ¯ M. We also explore some connections between optimal transportation and spacelike Lagrangian submanifolds in symplectic geometry. Using the pseudoRiemannian structure, we extend Ma, Trudinger and Wang’s conditions to transportation costs on differentiable manifolds, and provide a direct elementary proof of a maximum principal characterizing it due to Loeper, relaxing his hypotheses even for subdomains M and ¯ M of Rn. This maximum principle plays a key role in Loeper’s Hölder continuity theory of optimal maps. Our proof allows his theory to be made logically independent of all earlier works, and sets the stage for extending it to new global settings, such as general submersions and tensor products of the specific Riemannian manifolds he considered.
Cohomology of compact hyperkähler manifolds
, 1995
"... Let M be a compact simply connected hyperkähler (or holomorphically symplectic) manifold, dim H 2 (M) = n. Assume that M is not a product of hyperkaehler manifolds. We prove that the Lie group ..."
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Cited by 68 (16 self)
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Let M be a compact simply connected hyperkähler (or holomorphically symplectic) manifold, dim H 2 (M) = n. Assume that M is not a product of hyperkaehler manifolds. We prove that the Lie group