Study mean curvature forms on Kähler foliations with restrictions on basic cohomology.
problem Properties of mean curvature forms on transverse Kähler foliations.
method Analysis of mean curvature one-forms and their holomorphic/antiholomorphic counterparts.
result Restrictions on basic cohomology for automorphic mean curvature foliations.
Study on Dolbeault cohomology and Kähler foliations with new formulas and restrictions.
problem Investigating Dolbeault cohomology on transversely Kähler foliations.
method Developed new Weitzenböck formulas and conditions on curvature.
result Proved absence of nonzero basic-harmonic forms on foliations with positive Ricci curvature.
Study the structure of Kähler foliations with negative Ricci curvature.
problem Characterize the structure of Kähler foliations with negative Ricci curvature.
method Prove a de Rham type theorem decomposition on the leaf space.
result Characterize each factor in the decomposition of the leaf space.
Study on twisted Dolbeault cohomology in Kähler foliations.
problem Exploring cohomology in transverse Kähler foliations.
method Analysis of twisted basic Dolbeault cohomology and transverse hard Lefschetz theorem.
result Proved Kodaira-Serre type duality for twisted basic Dolbeault cohomology.
Classifies Kähler structures with special foliations using symplectic techniques.
problem Classifying Kähler structures with specific foliations.
method Weinstein's method of constructing symplectic bundles applied to Kähler data.
result New Kähler structures and metrics are obtained.
Classifies foliations on open Kähler manifolds with explicit curvature control.
problem Classifying foliations on open Kähler manifolds.
method Study of infinitesimal model associated to canonical connection for explicit curvature control.
result Complete classification of foliations on irreducible Hermitian symmetric spaces of compact type.
New proof of Aubin-Yau theorem for complex non-Kähler manifolds.
problem Transversally Kähler foliations as a generalization of Kähler manifolds.
method Adapting classical Aubin-Yau methods to transversally Kähler case under homological orientability condition.
result New, simpler proof of Vaisman Aubin-Yau theorem.
Study L2-transverse conformal Killing forms on foliated manifolds.
problem Prove vanishing theorems for L2-transverse conformal Killing forms.
method Analyzes L2-transverse conformal Killing forms on complete foliated Riemannian manifolds and Kahler foliations.
result Proves vanishing theorems for L2-transverse conformal Killing forms.
The aim of this paper is to describe complex foliations on Kahler surfaces.
The study introduces new foliations and structures on complex manifolds.
problem Understanding transverse Kähler structures on complex manifolds.
method Introducing holomorphic foliations and developing differential graded and bigraded algebras.
result Obtains quasi-isomorphic complexes to de Rham and Dolbeault complexes, similar to compact Kähler manifolds.
In this paper, we prove Kirchberg inequalities for any kahler spin foliations. Their limiting cases are then characterized as being transversal minimal Einstein foliations. The key point is to introduce the transversal kahlerian twistor operators.
Study numerically flat foliations on Kähler manifolds, proving new splitting theorems.
problem Understanding foliations with numerically flat tangent bundles on Kähler manifolds.
method Analyzing the structure of foliations on compact Kähler manifolds, extending earlier results.
result Smooth foliations with numerically flat tangent bundles induce a decomposition of the ambient manifold's tangent bundle.
On a closed, connected Riemannian manifold with a Kähler foliation of codimension q=2m, any transverse Killing r (≥2)-form is parallel (S. D. Jung and M. J. Jung [\ref{JJ2}], Bull. Korean Math. Soc. 49 (2012)). In this paper, we study transverse conformal Killing forms on Kähler foliations and prove that if th…
The aim of this paper is to classify compact, simply connected Kähler manifolds which admit totally geodesic, holomorphic complex homothetic foliation by curves.
Adapts Frolicher-type inequalities to foliations.
problem Determining foliations that can be made transversely Kahler.
method Adapting Frolicher-type inequalities to transversely holomorphic and symplectic foliations.
result Main results extend to orbifolds.
Riemann Poisson manifolds were introduced by the author in [1] and studied in more details in [2]. Kähler-Riemann foliations form an interesting subset of the Riemannian foliations with remarkable properties (see [3]). In this paper we will show that to give a regular Riemann Poisson structure on a manifold M is equi…
The study classifies natural almost Hermitian structures on specific Lie groups.
problem Classifying natural almost Hermitian structures on conformally foliated Lie groups.
method Examining 4-dimensional Riemannian Lie groups with a 2-dimensional conformal foliation and minimal leaves, constructing new examples of multi-dimensional structures.
result Constructing several new multi-dimensional examples of almost Kähler, integrable, and Kähler structures.
New theorems on Hodge numbers and Kähler structures derived from complex differential forms.
problem Deformation invariance and local stability of Hodge numbers and Kähler structures.
method Using the exponential operator and power series method, the approach focuses on d-closed extensions and foliated cases. result Local stabilities of transversely p-Kähler structures and new theorems on Hodge numbers. The paper classifies natural almost Hermitian structures on Lie groups with minimal conformal leaves.
problem Classifying natural almost Hermitian structures on Lie groups with minimal conformal leaves.
method Analyzing Lie groups with a 2-dimensional conformal foliation and classifying structures based on Lie algebra properties.
result 16 multi-dimensional almost Kähler families, 18 integrable families, and 11 Kähler families were constructed.
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.
In this article we study isometric immersions of nearly Kähler manifolds into a space form (specially Euclidean space) and show that every nearly Kähler submanifold of a space form has a totally umbilic foliation whose leafs are 6-dimensional nearly Kähler manifolds. Moreover using this foliation we show that there is …
Formality of foliation minimal model proved for co-Kähler manifolds.
problem Formality of foliation minimal model for co-Kähler manifolds.
method Demonstrated existence of parallel forms leading to simpler computation of cohomology.
result Formality of foliation minimal model proved for co-Kähler manifolds.
Study on Clairaut maps from nearly Kahler to Riemannian manifolds.
problem Characterizing Clairaut maps from nearly Kahler manifolds.
method Analyzing conditions for Clairaut maps to be totally geodesic foliations.
result Non-trivial examples of Clairaut maps are provided.
Let F be a Kähler foliation on a compact Riemannian manifold M. we study the properties of infinitesimal automorphisms on (M,F), and in particular we concentrate on the transversal conformal field, transversal projective field and transversally holomorphic field
We study the geometry of the leaf closure space of regular and singular Riemannian foliations. We give conditions which assure that this leaf space is a singular symplectic or Kähler space.
We study Riemannian foliations whose transverse Levi-Civita connection ∇ has special holonomy. In particular, we focus on the case where Hol(∇) is contained either in SU(n) or in Sp(n). We prove a Weitzenbock formula involving complex basic forms on Kähler foliations and we apply this formula for pointing…
In this Note we establish a relation between sections in globally generated holomorphic vector bundles on Kähler manifolds, isotropic with respect to a non-degenerate quadratic form, and totally geodesic foliations on Euclidean open domains. We find a geometric condition for a totally geodesic foliation to originate in…
In this paper, we study stability for harmonic foliations on locally conformal Kähler manifolds with complex leaves. We also discuss instability for harmonic foliations on compact submanifolds immersed in Euclidean spaces and compact homogeneous spaces.
The paper studies maps from pseudo-Hermitian to Kähler manifolds, proving harmonic map properties.
problem Analyzing maps between pseudo-Hermitian and Kähler manifolds.
method Investigates partial energy functionals and critical maps, proving foliated results for ∂b- and ∂b-harmonic maps. result Generalizes Siu's holomorphicity result to ∂b- and ∂b-harmonic maps. The space of broken hyperbolic structures generalizes the Teichmüller space of a punctured surface, and the space of projectivized broken measured foliations (equivalently, the space of projectivized affine foliations) generalizes the space of projectivized measured foliations. Just as projectivized measured foliations…
We construct real polarizable Hodge structures on the reduced leafwise cohomology of Kähler-Riemann foliations by complex manifolds. As in the classical case one obtains a hard Lefschetz theorem for this cohomology. Serre's Kählerian analogue of the Weil conjectures carries over as well. Generalizing a construction of …
Study shows rigidity for spin^c manifolds with foliated boundaries.
problem Rigidity of Riemannian spin^c manifolds with foliated boundaries.
method Integral inequality based on geometric quantities.
result Characterization of equality cases for specific manifolds.
This paper constructs hyper-Kähler metrics and hyper-Lagrangian foliations for a class of complex manifolds.
problem Constructing hyper-Kähler metrics and foliations for complex manifolds.
method Using isomonodromy flows and reductions of Plebański's heavenly equations, the paper constructs hyper-Kähler metrics and hyper-Lagrangian foliations.
result Explicit expressions for hyper-Kähler metrics and foliations are derived.
The purpose of this paper is to study the canonical foliations of a quaternion CR-submanifold of a quaternion Kähler manifold.
The purpose of this paper is to establish a completely new partial regularity theory on certain homogeneous complex Monge-Ampere equations. Our partial regularity theory will be obtained by studying foliations by holomorphic curves and and their relations to homogeneous complex Monge-Ampere equations. As applications, …
Study classifies 8D Lie groups with specific foliations and Hermitian structures.
problem Classifying 8D Lie groups with specific conformal and minimal foliations.
method Investigates left-invariant Hermitian structures on SU(2)×SU(2) leaves of 8D Lie groups. result Classifies Lie groups based on the integrability and types of Hermitian structures.
Decomposes Q-Fano Kähler-Einstein varieties into simpler components.
problem Understanding the structure of Q-Fano Kähler-Einstein varieties.
method Proves decomposition theorem using algebraically integrable foliations and stability conditions.
result Q-Fano Kähler-Einstein varieties decompose into simpler components.
Develops moment map theory for twisted scalar curvature in Kähler geometry.
problem Understanding the geometry of holomorphic submersions and foliations.
method Introduces a coupled system of equations on a holomorphic submersion.
result The coupled system appears as a moment map, generalizing to foliations.
Study on pointwise pseudo-slant warped product submanifolds in Kähler manifolds.
problem Characterizing and studying properties of pointwise pseudo-slant submanifolds in Kähler manifolds.
method Deriving conditions for integrability, totally geodesic foliation, and isometric immersions of pointwise pseudo-slant submanifolds.
result Necessary and sufficient conditions for pointwise pseudo-slant warped product and locally Riemannian product.
The purpose of this paper is to study the canonical totally real foliations of CR-submanifolds in a locally conformal Kähler manifold.
Paper introduces a new type of canonical metric for varieties with intermediate Kodaira dimension.
problem Finding canonical metrics for varieties with intermediate Kodaira dimension.
method Introducing a new notion of canonical metric and proving conditions for existence of relative Kähler-Einstein metric.
result Established conditions for the existence of relative Kähler-Einstein metric.
We obtain a local classification of complex homothetic foliations on Kaehler manifolds by complex curves. This is used to construct almost Kaehler, Ricci-flat metrics subject to additional curvature properties.
Study cohomology of quaternionic foliations and orbifolds.
problem Understanding cohomology of quaternionic foliations and orbifolds.
method Definition and proof of foliated versions of classical results for quaternionic Kähler manifolds.
result Formulation and proof of foliated versions of classical results for quaternionic Kähler manifolds.
Study on geometric properties of h-conformal semi-invariant submersions.
problem Exploring geometric characteristics of h-conformal semi-invariant submersions.
method Investigation of quaternionic Kähler manifolds and Riemannian manifolds, focusing on integrability of distributions and foliations.
result Established necessary and sufficient conditions for total geodesic submersions and twisted product manifolds.
The paper finds Kähler-Einstein metrics on certain degenerations.
problem Finding Kähler-Einstein metrics on degenerate spaces.
method Introducing fiberwise Kähler-Einstein foliation and solving relative CMA equation.
result Existence of relative Kähler-Einstein metric under K-poly stability conditions.
Study classifies quasi-Sasakian 3-manifolds, linking them to Sasakian and co-Kähler structures.
problem Classifying quasi-Sasakian structures in 3-manifolds.
method Deformation-based approach examining foliation and basic cohomology.
result Every quasi-Sasakian structure in this setting can be deformed into a Sasakian or co-Kähler one.
We study the foliation space of complex and invariant (by torsion of intrinsic Hermitian connection) umbilic distribution on an isometric immersion from a nearly Kähler manifold M into the Euclidean space. Under suitable conditions this leaf space is nearly Kähler and M can be decomposed into a product of this leaf…
In 1991, Dajczer and Rodriguez proved in [10] that a complete minimal real Kahler submanifold of codimension 2, if with complex dimension > 2, would be either holomorphic, or a cylinder, or complex ruled. In this article, we generalize their result to real analytic complete real Kahler submanifolds of codimension 4. Th…