Classification of toric self-dual Einstein gravitational instantons with negative cosmological constant
problem Classification of toric self-dual Einstein gravitational instantons
method Proving that if the conformal Kähler structure associated to one of the torus Killing fields is global and extends to an ALE manifold with no additional fixed points, then the corresponding self-dual Einstein instanton is precisely given by the infinite class of multipole solutions
result The classification of toric self-dual Einstein gravitational instantons is precisely given by the infinite class of multipole solutions
In this short note we prove that any complete four dimensional anti-self-dual (or self-dual) quasi-Einstein manifolds is either Einstein or locally conformally flat. This generalizes a recent result of X. Chen and Y. Wang.
Classifies Heisenberg-invariant self-dual Einstein manifolds with explicit metrics.
problem Classifying self-dual Einstein manifolds invariant under Heisenberg group actions.
method Explicit construction of metrics and analysis of completeness.
result Einstein constants can vary and solutions exist for non-zero Ricci curvature.
Proves Kähler-Einstein property for certain Einstein 4-manifolds.
problem Characterizing Einstein 4-manifolds with specific curvature conditions.
method Analyzes self-dual Weyl tensor and scalar curvature conditions.
result Einstein 4-manifolds are either anti-self-dual or Kähler-Einstein under certain conditions.
We provide a local classification of self-dual Einstein Riemannian four manifolds admitting a positively oriented Hermitian structure and characterize those which carry a hyperhermitian, non-hyperkählerian structure compatible with the negative orientation. We finally show that self-dual Einstein 4-manifolds obtained a…
We prove that simply connected Einstein four-manifolds of positive scalar curvature are conformally Kähler if and only if the determinant of the self-dual Weyl curvature is positive.
Superminimal surfaces in certain Einstein manifolds have a Calabi-Yau property.
problem Characterizing superminimal surfaces in specific Einstein manifolds.
method Utilizing twistor spaces and properties of holomorphic Legendrian curves.
result Superminimal surfaces in self-dual or anti-self-dual Einstein four-manifolds can be uniformly approximated by complete superminimal surfaces.
Study shows Einstein structures on 4-manifolds are rigid.
problem Rigidity of Einstein structures in four dimensions.
method Examined deformations of the round four-sphere and analyzed self-dual structure of Einstein manifolds.
result Any deviation from the standard metric of the round four-sphere breaks the Einstein condition.
We present a construction of complete self-dual Einstein metrics of negative scalar curvature on an uncountable family of manifolds of infinite topological type, which are enumerated by continued fraction expansions of irrational numbers. These manifolds may be regarded as limits of the resolutions of cyclic quotient s…
Study proves rigidity and gap theorems for specific metrics.
problem Existence and properties of self-dual and even Poincaré-Einstein metrics in 4D.
method Rigorous mathematical proofs, including gap theorems and rigidity results.
result Obtained new scalar conformal invariants and identified obstructions to metric existence.
The twistor space of self-dual positive Einstein manifolds naturally admits two 1-parameter families of Riemannian metrics, one is the family of canonical deformation metrics and the other is the family introduced by B. Chow and D. Yang in 1989. The purpose of this paper is to compare these two families. In particular …
The paper characterizes Einstein 4-manifolds with semi-definite curvature and derives inequalities.
problem Characterizing Einstein 4-manifolds with semi-definite sectional curvature.
method Using pointwise inequalities involving scalar curvature and Weyl curvatures.
result Closed 4-dimensional Einstein metrics saturating the pointwise inequality are completely characterized.
The author has elsewhere given a complete classification of those compact oriented Einstein 4-manifolds on which the self-dual Weyl curvature is everywhere positive in the direction of some self-dual harmonic 2-form. In this article, similar results are obtained when the self-dual Weyl curvature is everywhere non-negat…
Study pinches Weyl curvature on 4-manifolds, proving anti-self-duality.
problem Understanding Weyl curvature pinching on 4-manifolds.
method Analyzing harmonic and pinched self-dual Weyl curvature, proving anti-self-duality.
result Proves anti-self-duality for compact 4-manifolds with pinched self-dual Weyl curvature.
Affine manifolds linked to integrable equations and geometric structures.
problem Understanding the geometric and algebraic properties of affine manifolds.
method Analyzing the Kahlerian tangent bundle and multi-dimensional consistency of the TED equation.
result Affine manifolds are related to self-dual Einstein spaces and Hessian structures.
This paper is concerned with the construction of special metrics on non-compact 4-manifolds which arise as resolutions of complex orbifold singularities. Our study is close in spirit to the construction of the hyperkaehler gravitational instantons, but we focus on a different class of singularities. We show that any re…
In [29], Plebanski reformulated the anti-self-dual Einstein equations with non-zero scalar curvature as a first order PDE for a connection in an SO(3)-bundle over the four-manifold. The aim of this article is to place this differential equation in a new framework, in which it is both elliptic and a stationary point of …
Einstein 4-manifolds with negative self-dual curvature are locally rigid.
problem Conditions for local rigidity of Einstein 4-manifolds.
method New variational description of Einstein 4-manifolds and analysis of the Hessian of the poure connection action.
result Local rigidity of Einstein 4-manifolds with negative self-dual curvature.
If M is the underlying smooth oriented 4-manifold of a Del Pezzo surface, we consider the set of Riemannian metrics h on M such that W+(ω,ω)>0, where W+ is the self-dual Weyl curvature of h, and ω is a non-trivial self-dual harmonic 2-form on (M,h). While this open region in the space of Riemann…
The paper studies Riemannian four-manifolds and their twistor spaces using a moving frame approach.
problem Understanding the twistor spaces of Riemannian four-manifolds.
method Using the moving frame approach to analyze the twistor space Z of an oriented Riemannian four-manifold M. result Proves that first-order linear conditions on the almost complex structures of Z force the manifold M to be self-dual, and shows that the Atiyah-Hitchin-Singer twistor space bears a resemblance to a nearly Kähler manifold under first-order quadratic conditions. Study generalizes Yang-Mills equations for special complex surfaces.
problem Deriving equations for self-dual Yang-Mills fields on complex surfaces.
method Generalization of flat space Yang's and Newman's equations to conformally Kahler Riemannian 4-manifolds.
result Continuous group of hidden symmetries found only for conformally half-flat geometry.
Einstein 4-manifolds become conformally Kähler with positive scalar curvature.
problem Characterizing Einstein 4-manifolds with specific curvature properties.
method Combining LeBrun's conformal normalization with weighted divergence equations and first-order identities.
result Einstein metrics with simple largest eigenvalue of self-dual Weyl curvature become conformally Kähler with positive scalar curvature.
We use the quaternion Kahler reduction technique to study old and new self-dual Einstein metrics of negative scalar curvature with at least a two-dimensional isometry group, and relate the quotient construction to the hyperbolic eigenfunction Ansatz. We focus in particular on the (semi-)quaternion Kahler quotients of (…
In this paper, we study closed four-dimensional manifolds. In particular, we show that under various new pinching curvature conditions (for example, the sectional curvature is no more than 5/6 of the smallest Ricci eigenvalue) then the manifold is definite. If restricting to a metric with harmonic Weyl tensor, then it …
We simplify Hitchin's description of SU(2)-invariant self-dual Einstein metrics, making use of the tau-function of related four-pole Schlesinger system.
New proof shows certain 4D metrics are non-degenerate if curvature is negative definite.
problem Proving non-degeneracy of Poincaré-Einstein metrics.
method Proved non-degeneracy for 4D metrics satisfying a chiral curvature inequality.
result 4D Poincaré-Einstein metrics are non-degenerate if curvature is negative definite.
Using the twistor correspondence, we give a classification of toric anti-self-dual Einstein metrics: each such metric is essentially determined by an odd holomorphic function. This explains how the Einstein metrics fit into the classification of general toric anti-self-dual metrics given in an earlier paper (math.DG/06…
By refining Matsumoto's construction of Einstein ACH metrics, we construct a one parameter family of ACH metrics which solve the Einstein equation to infinite order and have a given three dimensional CR structure at infinity. When the parameter is 0, the metric is self-dual to infinite order. As an application, we give…
All anti-self-dual Einstein metrics with non-zero cosmological constant arise from a single second-order PDE.
problem Anti-self-dual Einstein metrics with non-zero cosmological constant
method A second-order PDE introduced by Lipstein and Nagy
result All such metrics arise from this equation
We present an analytic construction of complete non-compact 8-dimensional Ricci-flat manifolds with holonomy Spin(7). The construction relies on the study of the adiabatic limit of metrics with holonomy Spin(7) on principal Seifert circle bundles over asymptotically conical G2 orbifolds. The metrics we produce have an …
A local classification of locally conformal flat Riemannian Einstein-like four-manifolds as well as a local classification of all locally conformal flat Riemannian four-manifolds for which all Jacobi operators have parallel eigenspaces along every geodesic is given. Non-trivial explicit examples are presented. The prob…
We construct a new family of compact orbifolds with a positive self dual Einstein metric and a one-dimensional group of isometries. Together with another known family, these examples classify all 4-dimensional orbifolds that are quaternion Kaehler quotients by a torus of real Grassmannians.
Peng Wu recently announced a beautiful characterization of conformally Kaehler, Einstein metrics of positive scalar curvature on compact oriented 4-manifolds via the condition det (W^+) > 0. In this note, we buttress his claim by providing an entirely different proof of his result. We then present further consequences …
A family of new twistor string theories is constructed and shown to be free from world-sheet anomalies. The spectra in space-time are calculated and shown to give Einstein supergravities with second order field equations instead of the higher derivative conformal supergravities that arose from earlier twistor strings. …
Given a projective structure on a surface N, we show how to canonically construct a neutral signature Einstein metric with non-zero scalar curvature as well as a symplectic form on the total space M of a certain rank 2 affine bundle M→N. The Einstein metric has anti-self-dual conformal curvature and admits …
We find necessary and sufficient conditions for a Riemannian four-dimensional manifold (M,g) with anti-self-dual Weyl tensor to be locally conformal to a Ricci--flat manifold. These conditions are expressed as the vanishing of scalar and tensor conformal invariants. The invariants obstruct the existence of parallel …
In this paper we construct a family of examples of self-dual Einstain metrics of neutral signature, which are not Ricci flat, nor locally homogenous. Curvature of these manifolds is studied in details. These are obtained by the para-quaternionic reduction. We compare our examples with the orbifolds $\oo$ given by Galic…
Study para-Kähler-Einstein metrics and their non-integrable twistor distributions.
problem Characterize para-Kähler-Einstein metrics and their associated non-integrable twistor distributions.
method Use Cartan's method of equivalence and analyze the anti-self-dual Weyl tensor.
result Establish a correspondence between the anti-self-dual Weyl tensor and the Cartan quartic of the twistor distribution.
Minimal surfaces can be mapped to 3D with bounded images.
problem Mapping minimal surfaces to 3D with bounded images.
method Analyzes various types of minimal immersions into R3 and complex manifolds. result Every surface contains a Cantor set allowing bounded conformal minimal immersions.
We show the total space of the canonical line bundle L of a Kahler-Einstein manifold Xn supports integrable SU(n+1) structures, or Calabi-Yau structures. The canonical real line bundle L⊂L over a minimal Lagrangian submanifold M⊂X is calibrated in this setting and hence can …
New method solves 'googly problem' for Schwarzschild black holes.
problem Solving the 'googly problem' for non-chiral field configurations.
method Using twistor space and holomorphic loci to construct a non-self-dual, four-dimensional Kähler metric.
result First instance of a non-self-dual Einstein metric from holomorphic data in twistor space.
Study finds obstacles to solutions for specific equations on compact surfaces.
problem Existence of solutions to self-dual equations on compact surfaces.
method Depends on Higgs field zeroes and vortex number.
result Infinitely many Higgs fields for which solutions cannot exist.
The Goldberg-Sachs theorem is generalized for all four-dimensional manifolds endowed with torsion-free connection compatible with the metric, the treatment includes all signatures as well as complex manifolds. It is shown that when the Weyl tensor is algebraically special severe geometric restrictions are imposed. In p…
We give a new construction of Ricci-flat self-dual metrics which is a natural extension of the Gibbons--Hawking ansatz. We also give characterisations of both these constructions, and explain how they come from harmonic morphisms.
Constructing Einstein analogues with a non-zero cosmological constant
problem Constructing an Einstein analogue with a non-zero cosmological constant
method Proving the solution is either the Plebański-Demiański metric or has an anti-self-dual Weyl tensor
result For λ < 0, there is a conformal infinity separating two asymptotically hyperbolic metrics; one is globally conformal to an ALE scalar-flat Kähler metric; gravitational instantons with different topologies are constructed; the geometry is a 4-pole solution in the Calderbank-Pedersen classification
Functional for Spin(7) forms defined on compact manifolds.
problem Characterizing conformal Spin(7) forms on Riemannian manifolds.
method Algebraic equation of degree two for self-dual four-forms, construction of a functional.
result Construction of a functional whose critical points are conformally Ricci-flat Spin(7) structures.
The paper embeds CR manifolds into twistor spaces and constructs neutral hyperkähler metrics.
problem Embedding CR manifolds into twistor spaces and constructing neutral hyperkähler metrics.
method Embedding a real analytic twistor CR manifold into the twistor space of a Poincaré-Einstein metric, constructing the associated Fefferman ambient metric as a neutral hyperkähler metric.
result The construction of neutral hyperkähler metrics associated with twistor CR manifolds.
This review discusses solutions to Einstein's equations using twistor theory.
problem Finding solutions to Einstein's vacuum equations using twistor theory.
method Holomorphic vector bundles on twistor space and patching matrices.
result Holomorphic patching matrix P is simpler than the metric and determines the rod structure.