Explains hyperkähler manifolds and their properties.
problem None explicitly stated; focuses on introduction and properties.
method Elementary introduction and survey of properties.
result Properties of hyperkähler manifolds discussed.
Study on hyperKähler moment maps and hypertoric manifolds.
problem Classify hyperKähler manifolds with specific properties.
method Tri-Hamiltonian actions of tori, symplectic and hyperKähler geometry.
result HyperKähler moment maps have connected fibers and are surjective.
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.
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.
Study pseudo-hyperkähler geometry of curves in hyperkähler twistor spaces.
problem Understanding the geometry of rational curves in twistor spaces.
method Investigate pseudo-hyperkähler geometry of higher degree rational curves.
result Characterize the pseudo-hyperkähler structure of rational curves.
The study proves curvature bounds for hyperkähler manifolds.
problem Proving curvature invariants of hyperkähler manifolds.
method Analytical proof in complex dimension four, experimental proof in higher dimensions, verification for known manifolds.
result The conjectured curvature invariants are proven to be positive/negative for all known hyperkähler manifolds up to dimension eight.
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.
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 φα-sectional curvatures have constant holomorphic sectional curvatures in their HyperKahler contact distribution. Parabolic automorphisms on hyperkahler manifolds act ergodically on fibers.
problem Understanding the dynamics of parabolic automorphisms on hyperkahler manifolds.
method Analyzing the action of parabolic automorphisms on the second cohomology group and fibers of Lagrangian fibrations.
result Parabolic automorphisms preserving Lagrangian fibrations act ergodically on the fibers.
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.
Study on stability of hyperkähler flow in 4-manifolds.
problem Stability of hyperkähler flow in 4-manifolds.
method Extending results from mean curvature flow for minimal surfaces to hyperkähler flow.
result Obtained a dynamic stability theorem for hyperkähler flow.
New definition for a special type of complex manifolds.
problem Characterizing compact locally conformally hyperkähler manifolds.
method Equivalent definition using a nondegenerate complex two-form.
result Established a conformal analogue of Beauville's theorem.
New construction method for special geometric structures.
problem Creating strong HKT and generalized hyperkähler structures.
method Mapping tori construction of skew-symmetric torsion.
result Non-product structures with parallel skew-symmetric torsion.
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.
Essential dimension of hyperkähler manifolds families is at most 1.
problem Essential dimension of complex manifold families over compact bases.
method Proved essential dimension not greater than 1 for hyperkähler manifolds families over compact simply connected bases.
result Essential dimension of hyperkähler manifolds families is at most 1.
Holomorphic automorphisms on hyperkähler manifolds with high entropy are Kummer examples.
problem Characterizing holomorphic automorphisms with high entropy on hyperkähler manifolds.
method Using Jensen's inequality and properties of stable and unstable distributions, the authors show uniform contraction and expansion, leading to the conclusion that the manifold is birational to a torus quotient.
result Holomorphic automorphisms with high entropy on hyperkähler manifolds are Kummer examples.
A six-dimensional hyperkähler manifold has at most 23 second Betti number.
problem Bounding the second Betti number of hyperkähler manifolds.
method Assumption on Looijenga-Lunts-Verbitsky decomposition of cohomology.
result The second Betti number is at most 23.
Stratifies singular hyperkahler quotients into smooth manifolds.
problem Singular hyperkahler quotients by non-free actions.
method Topological stratification, global Poisson structures, local normal form.
result Quotients are locally isomorphic to linear complex-symplectic reductions.
This paper explores hyperkähler structures on specific orbits using different methods.
problem Investigate hyperkähler structures on adjoint orbits.
method Infinite dimensional hyperkähler reduction and symplectic geometry.
result Differences and connections between two methods are thoroughly investigated.
Proves C^2,alpha estimates for elliptic equations on hyperkähler manifolds.
problem Elliptic equations on hypercomplex manifolds.
method Proves C^2,alpha estimates under suitable assumptions.
result Solutions to specific elliptic equations on hyperkähler manifolds satisfy C^2,alpha estimates.
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.
Study generalizes Seiberg-Witten equations on hyperKahler manifolds.
problem Generalizing Seiberg-Witten equations on hyperKahler manifolds.
method Construction of a non-linear Dirac operator and derivation of transformation formula for the Dirac operator under conformal changes.
result Generalized Seiberg-Witten equations can be expressed as a second order PDE in terms of almost-complex structure and a conformal factor.
Study proves long-term solutions to a specific equation on hyperKähler manifolds.
problem Proving long-term existence and uniqueness of solutions to a parabolic quaternionic Monge-Ampère equation.
method Proved long-term existence and uniqueness using parabolic quaternionic Monge-Ampère type equation.
result Solution converges smoothly to the unique solution of the Monge-Ampère equation.
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.
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 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.
Solves long-time solutions for a specific equation on hyperkähler manifolds.
problem Finding solutions to a specific equation on hyperkähler manifolds.
method Introduced a parabolic quaternionic Monge-Ampère equation and proved its long-time solvability.
result Smooth convergence to a solution of the quaternionic Monge-Ampère equation.
Study of mean curvature flow in hyperkähler manifolds leading to complex Lagrangian submanifolds.
problem Understanding the behavior of mean curvature flow in hyperkähler manifolds.
method Definition of twistor energy and analysis of mean curvature flow starting from hyper-Lagrangian submanifolds.
result Mean curvature flow converges to a complex Lagrangian submanifold for sufficiently small twistor energy.
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.
We study relations between quaternionic Riemannian manifolds admitting different types of symmetries. We show that any hyperKahler manifold admitting hyperKahler potential and triholomorphic action of S^1 can be constructed from another hyperKahler manifold (of lower dimention) with an action of S^1 which fixes one com…
The paper proves a smooth Birman-Hilden theorem for hyperkähler manifolds.
problem Understanding the smooth mapping class groups of certain 4-manifolds.
method Geometric and Teichmüller-theoretic methods.
result Proves a global Torelli theorem for generalized Enriques manifolds.
Instantons on hyperkähler manifolds linked to holomorphic functions.
problem Understanding instantons on hyperkähler manifolds.
method Local bijection between gauge equivalence classes of instantons and holomorphic functions.
result Streamlined proof of Uhlenbeck's Compactness Theorem.
The paper studies mean curvature flow on hyperkähler manifolds and proves no Type 1 singularities.
problem Analyzing mean curvature flow on hyperkähler manifolds.
method Using optimal rigidity theorems and complex phase maps.
result No Type 1 singularities occur in mean curvature flow on hyperkähler manifolds.
We survey briefly the definition of the Rozansky-Witten invariants, and review their relevance to the study of compact hyperkahler manifolds. We consider how various generalisations of the invariants might prove useful for the study of non-compact hyperkahler manifolds, of quaternionic-Kahler manifolds, and of relation…
Study surjectivity of Kirwan map for generalized hyperkähler reduction.
problem Establishing surjectivity of Kirwan map for a specific class of Hamiltonian manifolds.
method Defined a close analogue of hyperkähler reduction for manifolds with equivariant functions under semi-linear G-actions. result Surjectivity of Kirwan map proved for the defined class of manifolds.
Study rigid classes on hyperkahler manifolds, showing general ones are rigid.
problem Characterize rigid classes on compact hyperkahler manifolds.
method Analyze eigenvectors of hyperbolic automorphisms and use BBF form.
result General parabolic classes on hyperkahler manifolds are rigid.
Let S be an infinite-dimensional manifold of all symplectic, or hyperkahler, structures on a compact manifold M, and Diff0 the connected component of its diffeomorphism group. The quotient $S/\Diff_0$ is called the Teichmuller space of symplectic (or hyperkahler) structures on M. MBM classes on a hyperkahler manifol…
The paper connects twist and modification methods to build hyperKähler structures.
problem Creating hyperKähler structures while preserving properties.
method Elementary deformations link twist and modification methods.
result Full classification of complete hyperKähler four-manifolds with tri-Hamiltonian symmetry.
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.
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…
Rozansky and Witten proposed a 3-dimensional sigma-model whose target space is a hyperkähler manifold. They conjectured that this theory has an associated TQFT, with Hilbert spaces given by certain cohomology groups of the hyperkähler manifold. On the other hand, there is a certain modified TQFT constructed by Murakami…
A weight system on graph homology was constructed by Rozansky and Witten using a compact hyperkähler manifold. A variation of this construction utilizing holomorphic vector bundles over the manifold gives a weight system on chord diagrams. We investigate these weights from the hyperkähler geometry point of view.
The Kobayashi pseudometric on a complex manifold is the maximal pseudometric such that any holomorphic map from the Poincaré disk to the manifold is distance-decreasing. Kobayashi has conjectured that this pseudometric vanishes on Calabi-Yau manifolds. Using ergodicity of complex structures, we prove this conjecture fo…
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.
Study of D2 ALF manifolds via hyperkahler quotients of affine spaces.
problem Resolving flat orbifold quotients of R4. method Infinite-dimensional generalization of Kronheimer's construction, hyperkahler quotients of affine spaces, singular equivariant instantons, stability of Nahm data.
result Construction of the family of D2 ALF manifolds as a deformation of the flat orbifold (R3×S1)/Z2. Let M be a compact irreducible hyperkahler manifold, from Bogomolov inequality [V1] we obtain forbidden values of the second Betti number b2 in arbitrary dimension. UPD: Unfortunately, decomposition of dual to BBF-form is not right in the main theorem. Instead of this work, take a look on recent preprints of Sawon…
Embeds cohomology of hyperkahler manifolds into torus cohomology.
problem Embedding cohomology of hyperkahler manifolds into torus cohomology.
method Kuga-Satake construction and embedding of graded cohomology spaces.
result Compatibility of embeddings with Hodge structures and Lie algebra actions.
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 ∇ˉ is defined. This metric connection is completely determined by HyperKahler contact distribution. We prove that HyperKahler conta…