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Selfdual Einstein metrics with torus symmetry
 J. Diff. Geom
"... Abstract. It is well known that any 4dimensional hyperkähler metric with two commuting Killing fields may be obtained explicitly, via the Gibbons–Hawking Ansatz, from a harmonic function invariant under a Killing field on R 3. In this paper, we find all selfdual Einstein metrics of nonzero scalar c ..."
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Cited by 63 (4 self)
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Abstract. It is well known that any 4dimensional hyperkähler metric with two commuting Killing fields may be obtained explicitly, via the Gibbons–Hawking Ansatz, from a harmonic function invariant under a Killing field on R 3. In this paper, we find all selfdual Einstein metrics of nonzero scalar curvature with two commuting Killing fields. They are given explicitly in terms of a local eigenfunction of the Laplacian on the hyperbolic plane. We discuss the relation of this construction to a class of selfdual spaces found by Joyce, and some Einstein–Weyl spaces found by Ward, and then show that certain ‘multipole ’ hyperbolic eigenfunctions yield explicit formulae for the quaternionkähler quotients of HP m−1 by an (m − 2)torus first studied by Galicki and Lawson. As a consequence we are able to place the wellknown cohomogeneity one metrics, the quaternionkähler quotients of HP 2 (and noncompact analogues), and the more recently studied selfdual Einstein Hermitian metrics in a unified framework, and give new complete examples. 1.
Selfdual spaces with complex structures, Einstein–Weyl geometry and geodesics
 Ann. Inst. Fourier
"... Abstract. We study the Jones and Tod correspondence between selfdual conformal 4manifolds with a conformal vector field and abelian monopoles on EinsteinWeyl 3manifolds, and prove that invariant complex structures correspond to shearfree geodesic congruences. Such congruences exist in abundance ..."
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Cited by 33 (7 self)
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Abstract. We study the Jones and Tod correspondence between selfdual conformal 4manifolds with a conformal vector field and abelian monopoles on EinsteinWeyl 3manifolds, and prove that invariant complex structures correspond to shearfree geodesic congruences. Such congruences exist in abundance and so provide a tool for constructing interesting selfdual geometries with symmetry, unifying the theories of scalarflat Kähler metrics and hypercomplex structures with symmetry. We also show that in the presence of such a congruence, the EinsteinWeyl equation is equivalent to a pair of coupled monopole equations, and we solve these equations in a special case. The new EinsteinWeyl spaces, which we call EinsteinWeyl “with a geodesic symmetry”, give rise to hypercomplex structures with two commuting triholomorphic vector fields. 1.
Mobius structures and two dimensional EinsteinWeyl geometry
 J. reine angew. Math
, 1997
"... this paper, a purely differential geometric point of view will be taken, the aim being to introduce two geometric structures that a conformal 2manifold might be equipped with, and to study the relationship between them. These structures are closely related to the projective and affine structures of ..."
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Cited by 22 (6 self)
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this paper, a purely differential geometric point of view will be taken, the aim being to introduce two geometric structures that a conformal 2manifold might be equipped with, and to study the relationship between them. These structures are closely related to the projective and affine structures of Riemann surface theory. The first structure can be viewed as a nonintegrable or nonholomorphic version of a complex projective structure, and will be called a Mobius structure. An integrable or flat Mobius structure on a conformal 2manifold induces a complex projective structure: the manifold possesses an atlas whose transition functions are complex Mobius transformations. However, contrary to common usage [9], the Mobius structures discussed herein are not necessarily integrable: they possess a curvature, analogous to the CottonYork tensor of a conformal 3manifold, whose vanishing is equivalent to integrability. Mobius structures are also different from real projective structures, in much the same way as conformal and real projective structures differ in higher dimensions. (In one dimension, Mobius and real projective structures do coincide and are always integrable.) The other topic of interest here is EinsteinWeyl geometry [3, 8, 14]. This is the geometry of a conformal manifold equipped with a compatible (or conformal) torsion free connection, such that the symmetric tracefree part of the Ricci tensor of this connection vanishes. These manifolds generalise Einstein manifolds in a natural way, and have been investigated in some detail recently (see [2, 6, 8] and references therein). In [11], Pedersen and Tod posed the problem of classifying compact two dimensional EinsteinWeyl manifoldsthe possible geometries of compact three dimensional EinsteinWeyl manifolds ha...
Explicit selfdual metrics on CP2# • • • #CP2
 J. Differential Geom
, 1991
"... We display explicit halfconformallyflat metrics on the connected sum of any number of copies of the complex projective plane. These metrics are obtained from magnetic monopoles in hyperbolic 3space by an analogue of the GibbonsHawking ansatz, and are conformal compactifications of asymptotically ..."
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Cited by 16 (1 self)
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We display explicit halfconformallyflat metrics on the connected sum of any number of copies of the complex projective plane. These metrics are obtained from magnetic monopoles in hyperbolic 3space by an analogue of the GibbonsHawking ansatz, and are conformal compactifications of asymptoticallyflat, scalarflat Kahler metrics on «fold blowups of C 2. The corresponding twistor spaces are also displayed explicitly, and are observed to be Moishezon manifolds — that is, they are bimeromorphic to projective varieties. 1.
Compact EinsteinWeyl Manifolds With Large Symmetry Group
 DUKE MATH. J
, 1995
"... A geometric classification of the compact fourdimensional EinsteinWeyl manifolds with at least fourdimensional symmetry group is given. Our results also sharpen previous results on fourdimensional Einstein metrics and correct Parker's topological classification of cohomogeneityone fourman ..."
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Cited by 16 (1 self)
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A geometric classification of the compact fourdimensional EinsteinWeyl manifolds with at least fourdimensional symmetry group is given. Our results also sharpen previous results on fourdimensional Einstein metrics and correct Parker's topological classification of cohomogeneityone fourmanifolds.
Gluing theorems for complete antiselfdual spaces
 Geom. Func. Analysis
"... 1.1. Summary. One of the special features of 4dimensional differential geometry is the existence of objects with selfdual (SD) or antiselfdual (ASD) curvature. The objects in question can be connections in an auxiliary bundle over a 4manifold, leading to the study of instantons in Yang–Mills th ..."
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Cited by 16 (5 self)
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1.1. Summary. One of the special features of 4dimensional differential geometry is the existence of objects with selfdual (SD) or antiselfdual (ASD) curvature. The objects in question can be connections in an auxiliary bundle over a 4manifold, leading to the study of instantons in Yang–Mills theory [DK91], or as in this paper, Riemannian metrics or conformal
Absolute objects and counterexamples: JonesGeroch dust, Torretti constant curvature, tetradspinor, and scalar density
 Studies in History and Philosophy of Modern Physics 37 (2006) forthcoming
"... James L. Anderson analyzed the novelty of Einstein’s theory of gravity as its lack of “absolute objects. ” Michael Friedman’s related work has been criticized by Roger Jones and Robert Geroch for implausibly admitting as absolute the timelike 4velocity field of dust in cosmological models in Einste ..."
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Cited by 15 (6 self)
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James L. Anderson analyzed the novelty of Einstein’s theory of gravity as its lack of “absolute objects. ” Michael Friedman’s related work has been criticized by Roger Jones and Robert Geroch for implausibly admitting as absolute the timelike 4velocity field of dust in cosmological models in Einstein’s theory. Using Nathan Rosen’s action principle, I complete Anna Maidens’s argument that the problem is not solved by prohibiting variation of absolute objects in an action principle. Recalling Anderson’s proscription of “irrelevant ” variables, I generalize that proscription to locally irrelevant variables that do no work in some places in some models. This move vindicates Friedman’s intuitions and removes the JonesGeroch counterexample: some regions of some models of gravity with dust are dustfree and so naturally lack a timelike 4velocity, so diffeomorphic equivalence to (1,0,0,0) is spoiled. Torretti’s example involving constant curvature spaces is shown to have an absolute object on Anderson’s analysis, viz., the conformal spatial metric density. The previously neglected threat of an absolute object from an orthonormal tetrad used for coupling spinors to gravity appears resolvable by eliminating irrelevant fields. However, given Anderson’s definition, GTR itself has an absolute object (as Robert Geroch has observed recently): a change of variables to a conformal metric density and a scalar density shows that the latter is absolute.
Integrable Background Geometries
"... Four geometric structures, in dimensions one to four, are studied and related. Each structure is governed by a nonlinear di#erential equation, and each solution of this equation determines a background geometry on which, for any Lie group G, a gauge theory is defined. In four dimensions, the geometr ..."
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Cited by 15 (1 self)
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Four geometric structures, in dimensions one to four, are studied and related. Each structure is governed by a nonlinear di#erential equation, and each solution of this equation determines a background geometry on which, for any Lie group G, a gauge theory is defined. In four dimensions, the geometry is selfdual conformal geometry and the gauge theory is selfdual YangMills theory, while the lower dimensional structures are reductions of this. Any solution of the gauge theory on any kdimensional geometry, such that the gauge group H acts transitively on an #manifold, determines a (k + #) dimensional geometry (k + # 6 4) fibering over the kdimensional geometry with H as a structure group. In the case of an #dimensional group H acting on itself by the regular representation, all (k + #)dimensional geometries with symmetry group H are locally obtained in this way. This framework unifies and extends known results about dimensional reductions of selfdual conformal geometry and the selfdual YangMills equation, and provides a rich supply of constructive methods. In one dimension, generalized Nahm equations provide a uniform geometric treatment of four pole isomonodromic deformation problems, and the SU(#) Toda equation is shown to be related to a generalized Nahm equation via a hodograph transformation. In two dimensions, the Di# (S 1 ) Hitchin equations are shown to be equivalent to the hyperCR EinsteinWeyl equations, while in three and four dimensions, the constructions of this paper help to organize the huge range of examples of EinsteinWeyl and selfdual spaces in the literature, as well as providing some new ones.
FIBRATIONS AND FUNDAMENTAL GROUPS OF KÄHLER–WEYL MANIFOLDS
, 811
"... ABSTRACT. We extend the Siu–Beauville theorem to a certain class of compact Kähler–Weyl manifolds, proving that they fiber holomorphically over hyperbolic Riemannian surfaces whenever they satisfy the necessary topological hypotheses. As applications we obtain restrictions on the fundamental groups ..."
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Cited by 15 (3 self)
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ABSTRACT. We extend the Siu–Beauville theorem to a certain class of compact Kähler–Weyl manifolds, proving that they fiber holomorphically over hyperbolic Riemannian surfaces whenever they satisfy the necessary topological hypotheses. As applications we obtain restrictions on the fundamental groups of such Kähler–Weyl manifolds, and show that in certain cases they are in fact Kähler. 1.
A Conformally Invariant Holographic Two–Point Function on the Berger Sphere
, 2004
"... We apply our previous work on Green’s functions for the four–dimensional quaternionic Taub–NUT manifold to obtain a scalar two–point function on the homogeneously squashed three–sphere (otherwise known as the Berger sphere), which lies at its conformal infinity. Using basic notions from conformal ge ..."
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Cited by 11 (0 self)
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We apply our previous work on Green’s functions for the four–dimensional quaternionic Taub–NUT manifold to obtain a scalar two–point function on the homogeneously squashed three–sphere (otherwise known as the Berger sphere), which lies at its conformal infinity. Using basic notions from conformal geometry and the theory of boundary value problems, in particular the Dirichlet–to–Robin operator, we establish that our two–point correlation function is conformally invariant and corresponds to a boundary operator of conformal dimension one. It is plausible that the methods we use could have more general