The paper extends Gray's result to quaternion-Kähler manifolds.
problem Understanding quaternion-Kähler manifolds with non-negative quaternionic sectional curvature.
method Introducing quaternionic sectional curvature, proving Wolf spaces have non-negative curvature, and using nearly Kähler twistor spaces.
result Every quaternion-Kähler manifold with non-negative quaternionic sectional curvature is a Wolf space.
New construction generalizes quaternionic Kähler/hyper-Kähler correspondence.
problem Constructing hypercomplex manifolds from quaternionic ones.
method Constructing hypercomplex manifold M′ from quaternionic manifold M with U(1)-symmetry. result Obtained a compact homogeneous hypercomplex manifold without hyper-Kähler structure.
Study eigenvalue estimates on Kähler and quaternion Kähler manifolds.
problem Estimating first eigenvalues in Kähler and quaternion Kähler manifolds.
method Using Kendall-Cranston coupling to analyze eigenvalues.
result Eigenvalue estimates in terms of dimension, diameter, and curvature.
Study on quaternionic bisectional curvature for quaternion-Kähler manifolds.
problem Characterize quaternionic bisectional curvature on quaternion-Kähler manifolds.
method Analyzing properties of quaternionic bisectional curvature on specific manifolds.
result Non-negative quaternionic bisectional curvature is only on quaternionic projective space.
Study eigenvalues of p-Laplacian on quaternionic Kähler manifolds.
problem Finding lower bounds for eigenvalues of p-Laplacian on quaternionic Kähler manifolds.
method Analytical proofs for both Neumann and Dirichlet boundary conditions.
result Established lower bounds for eigenvalues on compact quaternionic Kähler manifolds.
We analyze degenerate homogeneous structures of linear type in the pseudo-Kähler and para-Kähler cases. The local form and the holonomy of pseudo-Kähler or para-Kähler manifolds admitting such structure are obtained. In addition the associated homogeneous models are studied exhibiting their relation with the incomplete…
Study diameter bounds on Kähler and quaternionic Kähler manifolds with positive curvature.
problem Determine diameter bounds for Kähler and quaternionic Kähler manifolds under curvature positivity.
method Define orthogonal Bakry-Émery tensor, study diameter theorems, and derive Bonnet-Myers type bounds.
result Sharper diameter bounds than in Riemannian case under specific curvature assumptions.
Tight embedding proves Gromov norm for quaternionic Kähler class.
problem Calculating the Gromov norm for quaternionic Kähler class.
method Embedding of quaternionic hyperbolic disc into quaternionic hyperbolic n-space.
result Value of the Gromov norm of the quaternionic Kähler class determined.
Defines quaternionic k-vector fields on quaternionic Kähler manifolds.
problem No specific problem stated; focuses on definition and properties.
method Introduced a modified Dirac operator to define quaternionic k-vector fields.
result Calculated the dimension of quaternionic k-vector fields on HPn. We study the class of non-degenerate homogeneous structures of linear type in the pseudo-Kähler, para-Kähler, pseudo-quaternion Kähler and para-quaternion Kähler cases. We show that these structures characterize spaces of constant holomorphic, para-holomorphic, quaternion and para-quaternion sectional curvature respect…
The conformal infinity of a quaternionic-Kahler metric on a 4n-manifold with boundary is a codimension 3-distribution on the boundary called quaternionic contact. In dimensions 4n-1 greater than 7, a quaternionic contact structure is always the conformal infinity of a quaternionic-Kahler metric. On the contrary, in dim…
Local classification of quaternion-Kähler metrics with rotating S1-symmetry.
problem Classifying quaternion-Kähler metrics with specific symmetries.
method Quaternionic Feix--Kaledin construction and explicit construction of holomorphic contact distributions.
result Quaternion-Kähler metrics with rotating S1-symmetry are determined by a Kähler metric and a line bundle. Finite volume ends found in quaternionic Kähler manifolds.
problem Existence of quaternionic Kähler manifolds with finite volume ends.
method Proof in all dimensions 4m≥4. result Existence of quaternionic Kähler manifolds with finite volume ends.
New theorem links quaternionic-Kähler manifolds to symmetric spaces.
problem Understanding curvature properties of quaternionic-Kähler manifolds.
method Analyzing positive scalar curvature and nonnegative sectional curvature.
result Compact quaternionic-Kähler manifolds with these properties are symmetric.
Formulae for curvature of quaternionic Kähler manifolds derived from hyper-Kähler data.
problem Deriving curvature formulae for quaternionic Kähler manifolds.
method Using HK/QK correspondence and Levi-Civita connection, we express curvature in terms of hyper-Kähler data.
result Formulae for curvature of quaternionic Kähler manifolds, including norm of curvature tensor.
Construct quaternionic-Kähler metrics from special Kähler manifolds with specific BPS structure variations.
problem Construct quaternionic-Kähler metrics from special Kähler manifolds with mutually local variations of BPS structures.
method Construct quaternionic-Kähler metrics from a conical special Kähler manifold with a certain type of mutually-local variation of BPS structures. Provide global and local explicit formulas for the quaternionic-Kähler metric.
result Construct quaternionic-Kähler metrics that are positive-definite and deform the 1-loop corrected Ferrara-Sabharval metric.
The study proves completeness of certain Kähler manifolds and their families.
problem Completeness of projective special Kähler and quaternionic Kähler manifolds.
method Analyzing boundary behavior and cubic prepotential to prove completeness and define families of manifolds.
result Proves completeness of projective special Kähler manifolds with regular boundary behavior and cubic prepotential.
A quaternionic Kähler manifold M is called {\it positive} if it has positive scalar curvature. The main purpose of this paper is to prove several connectedness theorems for quaternionic immersions in a quaternionic Kähler manifold, e.g. the Barth-Lefschetz type connectedness theorem for quaternionic submanifolds in a p…
Study Brownian motions and heat kernel bounds on Kähler and quaternion Kähler manifolds.
problem Understanding Brownian motions and heat kernel bounds on specific geometric manifolds.
method Sharp Laplacian comparison theorems and Cheeger-Yau type lower bounds for heat kernels.
result Sharp Cheeger-Yau type lower bounds for heat kernels and Dirichlet eigenvalues of metric balls.
Quaternion-Kähler manifolds' stability and rigidity of scalar curvature studied.
problem Stability and rigidity of scalar curvature in quaternion-Kähler manifolds.
method Analysis of stability and rigidity conditions using Einstein manifold properties.
result Quaternion-Kähler manifolds of negative scalar curvature are stable and scalar curvature rigid.
Novel approach to instantons on quaternionic Kähler manifolds.
problem Understanding Yang-Mills instantons on quaternionic Kähler manifolds.
method Extension of harmonic space method for hyperkähler manifolds.
result Established a bijection between instantons and holomorphic maps.
Generalizes HK/QK correspondence to para-geometries.
problem Tackles the HK/QK correspondence in para-geometries.
method Provides a new proof and constructs deformations.
result Shows quaternionic Kahler target manifold for para-geometries.
Resolves gap problem for quaternion-Hermitian structures.
problem Determine maximal and submaximal symmetry dimensions for quaternion-Hermitian structures.
method Classifies structures with specific symmetry dimensions and studies geometric properties of submaximally symmetric spaces.
result Identifies locally conformally quaternion-Kähler and quaternion-Kähler with torsion structures.
We show that a para-quaternion nearly Kahler manifold is necessarily a para-quaternion Kahler manifold
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…
Paper constructs an example of a non-compact submanifold in a quaternionic Kähler symmetric space.
problem Tackles the construction of a non-compact totally complex submanifold in a quaternionic Kähler symmetric space.
method Uses an isometric action of a compact Lie group and a maximal totally geodesic sphere.
result Proves the existence of a non-compact totally complex submanifold of maximal dimension in a compact quaternionic Kähler symmetric space.
Study shows deformations of quaternionic Kähler manifolds are locally inhomogeneous.
problem Understanding deformations of quaternionic Kähler manifolds.
method Proved one-loop deformation of quaternionic Kähler manifolds are locally inhomogeneous.
result Full isometry group of one-loop deformations has cohomogeneity one.
We study hyperkahler cones and their corresponding quaternion-Kahler spaces. We present a classification of 4(n-1)-dimensional quaternion-Kahler spaces with n abelian quaternionic isometries, based on dualizing superconformal tensor multiplets. These manifolds characterize the geometry of the hypermultiplet sector of p…
New vanishing theorems for harmonic and pluriharmonic functions on Kähler and quaternionic Kähler manifolds.
problem Vanishing theorems for harmonic and pluriharmonic functions on Kähler and quaternionic Kähler manifolds.
method Utilized refined Kato type inequalities and Böchner technique to generalize results to Lp-integrable pluriharmonic functions and harmonic 1-forms. result Proved vanishing property of pluriharmonic functions with finite Lp energy on complete Kähler manifolds. We prove that a compact quaternionic-Kähler manifold of dimension 4n≥8 admitting a conformal-Killing 2-form which is not Killing, is isomorphic to the quaternionic projective space, with its standard quaternionic-Kähler structure.
Study on quaternionic Kähler manifolds and their integrable Hermitian structures.
problem Properties and integrability of quaternionic Kähler manifolds with Killing vector fields.
method Analysis of quaternionic Kähler manifolds with Killing vector fields, deriving conditions for integrability and conformal Kählerity.
result For a large class of quaternionic Kähler manifolds, the structure ildeJ1 is integrable. Classifies Lie group representations linked to quaternion-Kähler symmetric spaces.
problem Classifying representations of Lie groups with specific orbit spaces.
method Analyzing representations of Sp(1)k-extensions and quaternion-Kähler symmetric spaces. result Representations are derived from isotropy representations of quaternion-Kähler symmetric spaces.
In this paper, we give an optimal lower bound for the eigenvalues of the basic Dirac operator on a quaternion-Kahler foliation. The limiting case is characterized by the existence of quaternion-Kahler Killing spinors. We end this paper by giving some examples.
We describe explicitly all quaternionic contact hypersurfaces (qc-hypersurfaces) in the flat quaternion space $\Hnn$ and the quaternion projective space. We show that up to a quaternionic affine transformation a qc-hypersurface in $\Hnn$ is contained in one of the three qc-hyperquadrics in $\Hnn$. Moreover, we show tha…
Analogous to 4D, new Ansatz for quaternionic Kähler spaces with a free action.
problem Describing the geometry of quaternionic Kähler spaces with a free action.
method Gibbons-Hawking-like Ansatz based on quaternionic Kähler moment map.
result Explicit equivariant completion of twistor space construction.
The paper studies quaternionic Kähler manifolds and their fibration by solvmanifolds.
problem Geometry of quaternionic Kähler manifolds and their principal orbits.
method Analysis of cohomogeneity one examples and deformation of non-compact symmetric spaces.
result Principal orbits form a fibration by solvmanifolds under zero deformation parameter.
We give an explicit formula for the quaternionic Kähler metrics obtained by the HK/QK correspondence. As an application, we give a new proof of the fact that the Ferrara-Sabharwal metric as well as its one-loop deformation is quaternionic Kähler. A similar explicit formula is given for the analogous (K/K) correspondenc…
The paper studies quaternionic structures on GKM graphs and their relation to torus actions on quaternionic projective spaces.
problem Understanding quaternionic structures on GKM graphs and their implications for torus actions.
method Introducing quaternionic structures on GKM graphs and analyzing their properties in the context of torus actions.
result Abstract GKM graphs with specific 2-face structures correspond to torus actions on quaternionic projective spaces or Grassmannians.
Constructs quaternionic complexes over unimodular quaternionic manifolds.
problem Existence and properties of quaternionic complexes on unimodular quaternionic manifolds.
method Elementary calculation and conformal invariance.
result Ellipticity and Hodge-type decomposition of quaternionic complexes.
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.
Proves local existence of quaternionic contact structures using Cartan-Kähler theory.
problem Proving the existence of quaternionic contact structures locally.
method Utilized Cartan-Kähler theory to prove local existence.
result Depends on 2n+2 real analytic functions of 2n+3 variables modulo diffeomorphisms.
New submersions found in nearly Kähler geometry.
problem Exploring submersions in nearly Kähler geometry.
method Using reducible holonomy representations and connections with parallel skew torsion.
result Parallel 3-(α,δ)-Sasaki manifolds admit 1-dimensional submersions onto nearly Kähler orbifolds. Researchers create non-homogeneous finite-volume ends on quaternionic Kähler manifolds.
problem Constructing non-homogeneous quaternionic Kähler manifolds with finite volume ends.
method Using cohomogeneity one deformation of symmetric spaces, the researchers constructed manifolds with specific fundamental groups.
result The constructed manifolds are aspherical and have finite volume ends, not locally homogeneous.
Study eigenvalues on quaternion-Kähler manifolds with geometric bounds.
problem Estimating eigenvalues on quaternion-Kähler manifolds.
method Lower bounds derived from modulus of continuity estimates for heat equation solutions and Laplace comparison theorem.
result Established bounds for first nonzero eigenvalues in terms of dimension, diameter, and scalar curvature.
New spaces identified with specific properties.
problem Characterizing homogeneous spaces with quaternionic structures.
method Analyzing pseudo-Riemannian almost quaternionic homogeneous spaces with irreducible isotropy.
result Spaces are locally isometric to quaternionic Kähler symmetric spaces under certain conditions.
Geometric structures on quaternionic unit ball for slice regular Möbius transformations.
problem No new problem introduced.
method Introducing Hermitian, Riemannian, and Kähler-like structures on quaternionic unit ball using regular Möbius transformations.
result Geometric structures are natural generalizations of complex setup and solve problems not achieved by other geometries.
Study on stability of quaternion-Kähler manifolds using eigenvalue estimates.
problem Stability problem for positive quaternion-Kähler manifolds.
method Description of infinitesimal Einstein deformations and destabilising directions in terms of Laplace eigenfunctions and symmetric 2-tensors. Improved eigenvalue estimates for the Hodge-Laplacian on 2-forms.
result Sharp lower bound for the first non-zero eigenvalue on the parallel subbundle Sym^2 E of the 2-form bundle.
We show that a 1-parameter family Ricci flow ancient solutions arises from the natural collapsings of the twistor space of positive quaternion Kähler manifolds. We use these ancient solutions to show that a positive quaternion Kähler manifold is isometric to one of the Wolf spaces.