Study on stability of Sasaki structures under deformations.
problem Stability of Sasaki structures under transverse holomorphic deformations.
method Analysis of transverse Kähler holonomy groups and stability properties.
result Stability of ${\oldmathcal S}$ under certain conditions on Sasaki manifolds.
The abstract discusses transforming Vaisman metrics into Kähler-Einstein structures.
problem Transforming Vaisman metrics into Kähler-Einstein structures.
method Using the transverse Kähler-Ricci flow on the canonical foliation of a closed Vaisman manifold.
result Direct proof of short time existence of transverse Kähler-Ricci flow on Vaisman manifolds.
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. Extends pluripotential theory to Sasaki manifolds for cscs metrics.
problem Existence of Sasaki metrics with constant scalar curvature.
method Transverse Kahler structure to generalize Kahler pluripotential theory.
result Generalizes geometric structure of space of potentials in Sasaki setting.
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 a specific type of Lie algebras with Kähler and contact properties.
problem Characterizing and classifying transversely Kähler almost contact metric Lie algebras.
method Analyzing properties of Lie algebras with contact forms and Kähler structures, considering center dimensions and quotient properties.
result Classification of 5-dimensional η-Einstein transversely Kähler almost contact metric Lie algebras.
The paper explores families of almost complex structures and transverse (p,p)-forms.
problem Understanding and producing families of almost p-Kähler structures.
method Producing families of almost p-Kähler structures (Jt,Ωt) on $\C^3$, $\C^4$, and T6. result The almost complex structures Jt cannot be locally compatible with any symplectic form for teq0. 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.
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…
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.
In this paper we study compact Sasaki manifolds in view of transverse Kähler geometry and extend some results in Kähler geometry to Sasaki manifolds. In particular we define integral invariants which obstruct the existence of transverse Kähler metric with harmonic Chern forms. The integral invariant f1 for the first…
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 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.
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.
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
The study provides obstructions and examples for p-symplectic structures on complex manifolds.
problem Obstructing the existence of p-symplectic structures on compact complex manifolds. method Analyzing properties of (p,p)-transverse forms and 2p-forms. result Found obstructions and examples of p-symplectic structures on compact complex manifolds. The paper explores p-Kähler structures on Lie group quotients.
problem Existence of p-Kähler structures on compact quotients of Lie groups. method Analyzes p-Kähler structures on nilmanifolds and almost abelian solvmanifolds. result Proves that (n−2)-Kähler almost abelian solvmanifolds of complex dimension n≥3 are Kähler. Hodge numbers of Sasakian manifolds remain unchanged under deformations.
problem Invariance of Hodge numbers under deformations of Sasakian manifolds.
method Analysis of deformations of Sasakian structures and use of transversely elliptic operators.
result Hodge numbers are invariant under arbitrary deformations of the Sasakian structure.
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.
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…
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.
Study on complex variation of Hodge structures for non-Kähler manifolds.
problem Understanding complex variation of Hodge structures for non-Kähler manifolds.
method Analyzes holomorphic families of compact complex manifolds with specific cohomology properties.
result Period map is holomorphic and transversal under given conditions.
The paper explores p-Kähler structures on complex manifolds and their properties.
problem Addressing conjectures on specific types of complex manifolds and their structures.
method Analyzing (n−2)-Kähler nilmanifolds and holomorphically parallelizable nilmanifolds, deriving necessary conditions for smooth curves of p-Kähler structures, studying cohomology classes, and providing examples. result Derives necessary conditions for the existence of smooth curves of p-Kähler structures and provides examples of compact complex manifolds with p-Kähler structures. Study of coupled Sasaki-Einstein and solitons metrics.
problem Existence and properties of coupled Sasaki-Einstein and solitons metrics.
method Isomorphism between Lie algebra and space of coupled basic functions, use of coupled twisted Laplacians, reduction to Kähler-Einstein metrics, existence of toric coupled Sasaki-Einstein metrics.
result Existence and properties of coupled Sasaki-Einstein and solitons metrics, reduction to known cases when applicable.
Adapts PDE method to prove L∞ estimates for complex Hessian equations.
problem Proving L∞ estimates for complex Hessian equations on transverse Kähler manifolds. method Adapts PDE approach of Guo-Phong-Tong and Guo-Phong-Tong-Wang [17, 18].
result Obtains L∞ estimate for transverse complex Monge-Ampère equations. Study Sasaki manifolds' scalar curvature, proving existence under certain conditions.
problem Prove existence of Sasaki structures with transverse constant scalar curvature.
method Adapt Chen-Cheng's strategy to Sasaki setting, proving existence under reduced proper K-energy condition. result Existence of Sasaki structures with transverse constant scalar curvature under reduced proper K-energy condition. Study finds critical points of volume functionals on Sasaki manifolds.
problem Finding Kähler-Einstein metrics on Sasaki manifolds.
method Revisited moment polytopes, applied to volume minimization.
result Transverse coupled Kähler-Einstein metrics found as critical points.
New class of manifolds studied with special properties.
problem Characterizing and studying a new class of almost contact metric manifolds.
method Introducing anti-quasi-Sasakian manifolds and analyzing their properties.
result Anti-quasi-Sasakian manifolds with constant curvature are flat and cokähler.
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.
In this paper, we show the convexity of the image of a moment map on a transverse symplectic manifold equipped with a torus action under a certain condition. We also study properties of moment maps in the case of transverse Kähler manifolds. As an application, we give a positive answer to the conjecture posed by Cupit-…
Any strictly pseudoconvex domain in C2 carries a complete Kahler-Einstein metric, the Cheng-Yau metric, with ``conformal infinity'' the CR structure of the boundary. It is well known that not all CR structures on the 3-sphere arise in this way. In this paper, we study CR structures on the 3-sphere satisfying a differen…
The article constructs an invariant for foliations without holonomy-invariant transverse measure.
problem Constructing an invariant for foliations without holonomy-invariant transverse measure.
method Uniform exponential decay estimate for Seiberg-Witten equations on non-compact 4-manifolds with exact symplectic ends.
result Constructs an invariant for smooth foliations without holonomy-invariant transverse measure.
The paper studies CR geometry and introduces a new quotient method.
problem CR geometry in arbitrary codimension.
method Levi-Kahler quotient for constructing Kahler metrics from CR structures.
result Explicit descriptions and characterizations of Levi-Kahler quotients of toric CR manifolds, especially products of odd dimensional spheres.
Study transverse J-holomorphic curves linking nearly Kähler CP3 to minimal surfaces.
problem Understanding J-holomorphic curves in nearly Kähler CP3. method Introducing transverse J-holomorphic curves and establishing Bonnet-type theorems. result Classification of flat tori and construction of moment-type maps.
This paper is concerned with (transversally) Kähler foliations. We proved here that the holonomy pseudo-group's lie algebra is semi-simple unde negativity assumptions of the Ricci tensor.
We classify simply connected compact Sasaki manifolds of dimension 2n+1 with positive transverse bisectional curvature. In particular, the Kähler cone corresponding to such manifolds must be bi-holomorphic to $\C^{n+1}\backslash \{0\}$. As an application we recover the Mori-Siu-Yau theorem on the Frankel conjecture a…
We show that Tolman's example (of a six dimensional Hamiltonian T2-space with isolated fixed points and no compatible Kähler structure) can be constructed from the flag variety U(3)/U(1)3 by U(2)-equivariant symplectic surgery. This implies that Tolman's space has a ``transversal multiplicity-free'' action of $…
Equivalence found between certain Kahler and Sasaki metrics.
problem Understanding relationships between Kahler and Sasaki metrics.
method Establishing an equivalence between conformally Einstein-Maxwell Kahler 4-manifolds and extremal Kahler 4-manifolds with non-vanishing scalar curvature.
result New existence and non-existence results for extremal Sasaki metrics.
We study an odd-dimensional analogue of the Goldberg conjecture for compact Einstein almost Kähler manifolds. We give an explicit non-compact example of an Einstein almost cokähler manifold that is not cokähler. We prove that compact Einstein almost cokähler manifolds with non-negative ∗-scalar curvature are cokähler…
Study on harmonicity of complex structure on product of trans-Sasakian manifolds.
problem Investigating harmonicity of complex structure on product of trans-Sasakian manifolds.
method Analysis of Levi-Civita connection and conditions for harmonicity on product manifold.
result Conditions for harmonicity of complex structure on product manifold of trans-Sasakian manifolds.
We construct the moduli space of contact instantons, an analogue of Yang-Mills instantons defined for contact metric 5-manifolds and initiate the study of their structure. In the K-contact case we give sufficient conditions for smoothness of the moduli space away from reducible connections and show the dimension is…
Deformations of the Reeb flow of a Sasakian manifold as transversely Kähler flows may not admit compatible Sasakian metrics anymore. We show that the triviality of the (0,2)-component of the basic Euler class characterizes the existence of compatible Sasakian metrics for given small deformations of the Reeb flow as tra…
The paper explores bi-Lagrangian structures in Teichmüller theory.
problem Exploring geometric structures on manifolds and their applications in Teichmüller theory.
method Review and introduction of bi-Lagrangian structures, focusing on symplectic and Lagrangian foliations.
result Complexification of real-analytic Kähler manifolds has a natural complex bi-Lagrangian structure.
We show that Perelman's W-functional can be generalized to Sasaki-Ricci flow. When the basic first Chern class is positive, we prove a uniform bound on the scalar curvature, the diameter and a uniform C1 bound for the transverse Ricci potential along the Sasaki-Ricci flow, which generalizes Perelman's results Kahler…
Paper generalizes Schwarz lemma for harmonic maps between Riemannian manifolds.
problem Generalizing Schwarz lemma for harmonic maps.
method Using Bochner techniques and sub-Laplacian comparison theorem.
result Established a generalization of Schwarz lemma for transversally harmonic maps.
Study on non-Kähler Calabi-Yau geometries on 3-folds with constraints.
problem Characterizing non-Kähler Calabi-Yau geometries on threefolds.
method Symmetry reduction and scalar PDE for Kähler structure.
result Equality of Bott-Chern number hBC1,1≥2 with specific conditions. We present reformulation of Mathieu's result on representing cohomology classes of symplectic manifold with symplectically harmonic forms. We apply it to the case of foliated manifolds with transversally symplectic structure and to symplectic orbifolds. We obtain in particular that such representation is always possibl…
Defines pre-Kähler structures and their properties.
problem Geometry of Levi degenerate CR hypersurfaces.
method Introduces pre-Kähler structures, holomorphic degeneration, and finite-nondegeneracy.
result Symmetry algebra is finite-dimensional if and only if structure is finitely nondegenerate.