Promotes Poisson deformations to hyperkähler structures.
problem Deforming hyperkähler cone metrics.
method Twistor methods for universal Poisson deformations.
result Produces incomplete hyperkähler metrics with applications.
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48 results for “hyperKahler metric”
A hyperKähler potential is a function rho that is a Kähler potential for each complex structure compatible with the hyperKähler structure. Nilpotent orbits in a complex simple Lie algebra are known to carry hyperKähler metrics admitting such potentials. In this paper, we explicitly calculate the hyperKähler potential w…
Constructing Einstein metrics on bundles over hyperKähler manifolds.
problem Finding Einstein metrics on bundles over hyperKähler manifolds.
method Constructing quaternion-Kähler and hypercomplex structures, studying infinitesimal symmetries, and performing QK reduction.
result Producing several explicit quaternion-Kähler metrics and new proofs of correspondences.
New method recovers hyperkähler metrics from twistor models.
problem Constructing and recovering hyperkähler metrics from twistor models.
method Construction of principal twistor models and universality theorem.
result Universality theorem for recovering hyperkähler metrics from twistor spaces.
The paper constructs complex structures on hypersurfaces in hyperkahler manifolds.
problem No specific problem stated; focuses on construction of structures.
method Construction of complex metric structures on hypersurfaces in hyperkahler manifolds.
result The construction of complex structures on hypersurfaces in hyperkahler manifolds.
Let O be a nilpotent orbit in g^C where G is a compact, simple group and g=Lie(G). It is known that O carries a unique G-invariant hyperKähler metric admitting a hyperKähler potential compatible with the Kirillov-Kostant-Souriau symplectic form. In this work, the hyperKähler potential is explicitly calculated when O is…
New HyperKahler structure found for 3-contact distributions on Sasakian manifolds.
problem Finding a HyperKahler structure for 3-contact distributions on Sasakian manifolds.
method Defined a special metric connection and proved curvature properties.
result 3-Sasakian manifolds with constant -sectional curvatures have constant holomorphic sectional curvatures in their HyperKahler contact distribution.
Extends parabolic study to flat hyperkähler manifolds.
problem Study problems in hyperhermitian geometry.
method Extends elliptic approach to parabolic setting.
result Solves problems in hyperhermitian geometry.
Study elliptic equations on hyperhermitian manifolds with flat hyperkähler metric.
problem Solving elliptic equations on compact hyperhermitian manifolds.
method Adapting Székelyhidi's approach to the hypercomplex setting.
result Prove a priori estimates for solutions to elliptic equations.
The paper generalizes hyperkahler metrics near Lagrangian submanifolds.
problem Constructing hyperkahler structures near complex Lagrangian submanifolds.
method Generalization of Feix-Kaledin theorem and deformations of holomorphic symplectic structures.
result Hyperkahler structures can be constructed on symplectic realizations of holomorphic Poisson manifolds.
Study collapsing geometry of hyperkähler 4-manifolds and prove conjectures.
problem Understanding collapsing geometry of hyperkähler 4-manifolds.
method Investigation of collapsing geometry and proving conjectures.
result Proved two conjectures about collapsed limits and asymptotic behavior of hyperkähler 4-manifolds.
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.
Exponential decay observed between two metrics on a moduli space.
problem Analyzing the difference between two metrics on a moduli space.
method Proving exponential decay along generic rays in the Hitchin moduli space.
result Exponential decay of the difference between and .
In this paper, we study hyperkahler metric and practice GMN's construction of hyperkahler metric on focus-focus fibrations. We explicitly compute the action-angel coordinates on the local model of focus-focus fibration, and show its semi-global invariant should be harmonic to admit a compatible holomorphic 2-form. Then…
Study the space of hyperkähler metrics on a 4-manifold with boundary.
problem Understanding the space of hyperkähler triples on a 4-manifold with boundary.
method Prove the moduli space is a smooth manifold, describe tangent space, explore boundary value problem, study explicit examples.
result Prove the moduli space is a smooth infinite-dimensional manifold and describe the tangent space.
Compactifies metrics on K3 surfaces with algebraic description.
problem Classify Gromov-Hausdorff limits of K3 surfaces with fixed structures or polarizations.
method Algebraic description of Gromov-Hausdorff compactification.
result Classification of Gromov-Hausdorff limits of K3 surfaces.
Study of a basic Hitchin equation on Sasakian 3-folds, showing hyperKähler metric.
problem Understanding moduli spaces of the basic Hitchin equation on Sasakian 3-folds.
method Construction of moduli space and calculation of its dimension.
result Moduli space admits a hyperKähler metric.
In this paper, the HyperKahler contact distribution of a 3-Sasakian manifold is studied. To analyze the curvature properties of this distribution, the special metric connection is defined. This metric connection is completely determined by HyperKahler contact distribution. We prove that HyperKahler conta…
New approach to scalar curvature using hyperkähler reduction.
problem Finding solutions to scalar curvature equations.
method Explicit construction of hyperkähler metrics and moment map equations.
result Existence of solutions to moment map equations on ruled surfaces.
Researchers create a new metric on complex projective space bundles.
problem Constructing a hyperkähler metric on complex projective space bundles.
method Explicit construction in local coordinates, using holomorphic isomorphism to coadjoint orbits.
result A hyperkähler metric on twisted cotangent bundles of .
Study of a metric on cotangent bundle spaces of Kähler quotients.
problem Construct and analyze a metric on the total space of cotangent bundles to Kähler quotients.
method Use hyperkähler reduction to construct a hyperkähler metric, prove its coincidence with the Feix-Kaledin metric, and investigate its completeness.
result The metric completion of the space is a stratified hyperkähler space with an algebraic structure.
New collapsing metrics on K3 surface modeled on ALF gravitational instantons.
problem Constructing new collapsing hyperkähler metrics on K3 surface.
method Deforming approximate hyperkähler metrics by gluing ALF gravitational instantons to a background metric.
result The collapsing metrics are modeled by ALF gravitational instantons of dihedral and cyclic types.
We show that on Hilbert scheme of points on $\C^2$, the hyperkähler metric construsted by H. Nakajima via hyperkähler reduction is the Quasi-Asymptotically Locally Euclidean (QALE in short) metric constructed by D. Joyce.
Study shows rapid decay of Hitchin metric from semi-flat metric on Higgs bundles.
problem Analyzing the asymptotic behavior of the Hitchin metric on moduli spaces of Higgs bundles.
method Examined the decay rate of the difference between Hitchin and semi-flat metrics on smooth spectral curves.
result Exponential decay of the difference between Hitchin and semi-flat metrics as t approaches infinity.
Study shows how certain complex geometrical structures shrink to a simpler form.
problem Behavior of hyperkähler manifolds under specific conditions.
method Analyzes projective hyperkahler manifolds with holomorphic fibrations.
result Metrics of shrinking torus fibers collapse to a special Kahler manifold.
New hyperkähler metrics constructed via Riemann-Hilbert problems.
problem Constructing hyperkähler metrics on complex integrable systems with singular fibers.
method Solving Riemann-Hilbert problems and isomonodromic deformations to obtain Darboux coordinates.
result Holomorphic symplectic form constructed, leading to hyperkähler metric.
Solves a conjecture about hyperKähler manifolds using quaternionic Monge-Ampère equation.
problem Proving the solvability of a conjecture about hyperKähler manifolds with torsion.
method Solving quaternionic Monge-Ampère equation on hyperKähler manifolds without assuming flatness.
result Proves the conjecture for hyperKähler manifolds with trivial canonical bundle.
Study asymptotics of hyperkähler geometry on singular fibers of Hitchin moduli space.
problem Asymptotic hyperkähler geometry of -Hitchin moduli space over singular fibers.
method Extension of exponential convergence results to locally fiducial Higgs bundles and subintegrable systems.
result Hyperkähler metric converges exponentially to semi-flat metric on subintegrable systems.
Let M be a hyperkähler manifold. The S^2-family of complex structures compatible with the hyperkähler metric can be assembled into a single complex structure on Z=MxS^2; the resulting complex manifold is known as the twistor space of M. We describe the analogous construction for generalized complex structures in the se…
New symmetries and line bundles on hyperkähler spaces explored.
problem Exploring new symmetries on hyperkähler spaces.
method Introducing new continuous symmetries and their geometric implications.
result Found new types of symmetries and their associated line bundles.
The hyperKähler-quaternionic Kähler correspondence constructs quaternionic Kähler metrics from hyperKähler metrics with a rotating circle symmetry. We discuss how this may be interpreted as a combination of the twist construction with the concept of elementary deformation, surveying results of our forthcoming paper. We…
We review a construction of hyperkahler metrics proposed in joint work of Davide Gaiotto, Greg Moore and the author. A key ingredient in this construction is a collection of integer "DT invariants" obeying the wall-crossing formula of Kontsevich-Soibelman.
It is known that nilpotent orbits in a complex simple Lie algebra admit hyperKähler metrics with a single function that is a global potential for each of the Kähler structures (a hyperKähler potential). In an earlier paper the authors showed that nilpotent orbits in classical Lie algebras can be constructed as finite-d…
Sub-twistor metricsmath.DG
The paper studies metrics on hyperkähler manifolds using sub-twistor constraints.
problem Understanding metrics on hyperkähler manifolds with non-holonomic constraints.
method Introducing and analyzing sub-conic metrics, showing they are sub-Finsler.
result Explicit description of Finsler norms for sub-twistor metrics.
Moduli of K3 families over and complex-hyperkähler metrics induced by deformed twistor cyclesmath.AG
The paper solves a geometric problem related to K3 surfaces and complex-hyperkähler metrics.
problem Geometric meaning of small deformations of twistor cycles in K3 period domain.
method Construction of a moduli space for families of marked K3 surfaces and use of Penrose's Non-linear Graviton construction.
result Small deformations of twistor cycles induce complex-hyperkähler metrics on K3 surface families.
New minimal 2-spheres found in hyperkähler 4-manifolds, unstable and not holomorphic.
problem Characterizing stable minimal surfaces in hyperkähler 4-manifolds.
method Gluing construction using Scherk and Taub-NUT surfaces, harmonic map parametrization.
result Existence of unstable minimal 2-spheres with degree-1 Gauss lift, not holomorphic.
This paper connects complex hyperkähler structures to Donaldson-Thomas invariants.
problem Describing geometric structures on spaces of stability conditions.
method Introducing Joyce structures and showing their relation to complex hyperkähler structures.
result A Joyce structure on a complex manifold defines a complex hyperkähler structure on its tangent bundle.
Generalizes gravitational instanton metrics to a new class.
problem Finding explicit expressions for gravitational instantons.
method Quaternionic-based Ansatz generalizing Gibbons-Hawking Ansatz.
result Explicit expressions for type gravitational instantons.
The paper proves properties of a hyperkähler moduli space for surfaces with genus ≥ 2.
problem The hyperkähler moduli space of almost-Fuchsian structures on surfaces.
method Proofs using Teichmüller theory, Higgs bundles, and hyperbolic geometry.
result The hyperkähler structure on the moduli space extends the Weil-Petersson metric and Goldman symplectic structure.
Study fully nonlinear elliptic equations on compact hyperhermitian manifolds.
problem Solving fully nonlinear elliptic equations on compact hyperhermitian manifolds.
method Adapting Székelyhidi's approach to the hypercomplex setting, proving a priori estimates.
result Proves solvability of quaternionic Hessian and Monge-Ampère equations on compact flat hyperkähler manifolds.
Constructs hyperkähler metrics on Higgs bundle moduli spaces using Gaiotto coordinates.
problem Constructing hyperkähler metrics on moduli spaces of Higgs bundles.
method Using Gaiotto coordinates and solving Riemann-Hilbert problems, constructing a twistorial hyperkähler metric.
result The difference between the constructed hyperkähler metric and a simpler semiflat metric is exponentially suppressed.
The paper explores structures on 4-manifolds using hyperKahler geometry.
problem Exploring structures on 4-manifolds using hyperKahler geometry.
method Non-linear generalization of the Dirac operator, covariantly constant generalized spinor, harmonic spinor, generalized Seiberg-Witten equations.
result Existence of a covariantly constant, generalized spinor defines a Kahler structure on the base 4-dimensional manifold.
The Ooguri-Vafa space is linked to framed wild harmonic bundles via hyperkähler geometry.
problem Relating the Ooguri-Vafa space to Hitchin moduli spaces.
method Interpreting the Ooguri-Vafa space as framed wild harmonic bundles over CP1.
result The Ooguri-Vafa space is a moduli space of framed wild harmonic bundles.
Collapsing K3 surfaces, Tropical geometry and Moduli compactifications of Satake, Morgan-Shalen typemath.AG
Study Gromov-Hausdorff limits of K3 surface metrics via moduli compactification.
problem Understanding limits of K3 surface metrics.
method Moduli-theoretic framework for collapsing Ricci-flat Kahler metrics.
result Gromov-Hausdorff limits of hyperKahler metrics with fixed diameters.
Defines special Joyce structures for ASK manifolds encoding real HK structures.
problem Defining Joyce structures for ASK manifolds.
method Introducing special Joyce structures and showing they encode real HK structures.
result Special Joyce structures encode real hyperkähler structures on ASK manifolds.
The paper studies nilpotent structures in oriented neutral vector bundles and neutral hyperKähler structures.
problem Nilpotent structures in oriented neutral vector bundles and their relation to neutral hyperKähler structures.
method Defined -nilpotent structures for Lie subgroups of related to neutral hyperKähler structures.
result Existence of complex and paracomplex structures forming neutral hyperKähler structures if and only if there exists an -nilpotent structure.
Study describes metrics on moduli space of Higgs bundles, proving exponential decay rate.
problem Analyzing the geometry of moduli space of Higgs bundles.
method Perturbing from approximate solutions to describe metrics, comparing to semi-flat metrics.
result Proves exponential decay rate of .
A homogeneous Gibbons-Hawking ansatz is described, leading to 4-dimensional hyperkahler metrics with homotheties. In combination with Blaschke products on the unit disc in the complex plane, this ansatz allows one to construct infinite-dimensional families of such hyperkahler metrics that are, in a suitable sense, comp…