arXiv research

Study cohomology of quaternionic foliations and orbifolds.

We describe and construct here pseudo-Hermitian structures $θ$ without torsion (i.e. with transversal symmetry) whose Webster-Ricci curvature tensor is a constant multiple of the exterior differential $dθ$. We call these structures pseudo-Hermitian Einstein and our result states that they all can be derived locally fro…

New structures on symplectic manifolds derived from convex functions and matrices.

Study of toric generalized Kähler structures with strong Hamiltonian torus actions.

Abstract revisits Kähler reduction using GIT, generalizing results.

The paper proves conditions for Kähler manifolds with negative curvature.

For the sake of hyperk{ä}hler SYZ conjecture, finding holomorphic Lagrangian fibrations becomes an important issue. Toric hyperk{ä}hler manifolds are real dimension $4n$ non-compact hyperk{ä}hler manifolds which are quaternion analog of toric varieties. The $n$ dimensional residue circle action on it admitting a hyperk…

The paper explores metric reduction in generalized geometry and constructs holomorphic bundles.

Study properties of para-Kähler manifolds with conformal Einstein soliton metrics.

The paper studies Ricci curvature on Kähler-Ricci flow.

Toric hyperk{ä}hler manifolds are quaternion analog of toric varieties. Bielawski pointed out that they can be glued by cotangent bundles of toric varieties. Following his idea, viewing both toric varieties and toric hyperk{ä}her manifolds as GIT quotients, we first establish geometrical criteria for the semi-stable po…

Holomorphic Euler number vanishes for certain Kähler manifolds.

Almost contact structures can be identified with sections of a twistor bundle and this allows to define their harmonicity, as sections or maps. We consider the class of nearly cosymplectic almost contact structures on a Riemannian manifold and prove curvature identities which imply the harmonicity of their parametrizin…

We construct new explicit toric scalar-flat K{ä}hler ALE metrics on weighted projective spaces of non-compact type, which we use to obtain smooth extremal K{ä}hler metrics on appropriate resolutions of orbifolds. In particular, we obtain new extremal metrics certain resolutions of weighted projective spaces of compact …

Study complex Monge-Ampère flows on Kähler manifolds using Perron method.

In this note we prove the following result: There is a positive constant $ε(n,Λ)$ such that if $M^n$ is a simply connected compact K$\ddot{a}$hler manifold with sectional curvature bounded from above by $Λ$, diameter bounded from above by 1, and with holomorphic bisectional curvature $H \geq -ε(n,Λ)$, then $M^n$ is dif…

In this paper we prove that for a complete, connected and oriented Käler affine manifold $(M,G)$ of dimension $n,$ if it is Kähler affine Ricci flat or the K$\ddot{a}$hler affine scalar curvature $S\equiv0,$ ($n\leq 5$), then the universal covering manifold $\widetilde{M}$ of $M$ is isometric to the Euclidean n-space $…

Minimal hypersurfaces in nearly G2 manifolds have special properties.

Positive scalar curvature implies small 2-systoles in Kähler manifolds

Stability proven for complex equations on Kähler manifolds.

Develops theory for Kähler-Ricci flow on singular varieties.

The study proves compactness of Hamiltonian stationary Lagrangian surfaces in Kähler surfaces.

Study transverse Dolbeault cohomology for almost complex structures.

We investigate invariants of compact hyperk{ä}hler manifolds introduced by Rozansky and Witten: they associate an invariant to each graph homology class. It is obtained by using the graph to perform contractions on a power of the curvature tensor and then integrating the resulting scalar-valued function over the manifo…

Continuity of complex Monge-Ampère potentials on Kähler manifolds.

Study on symplectic Dirac operators on foliations, estimating eigenvalues.

Smooth maps in o-minimal structures are mostly transverse.

New transverse links solve contact manifold problems.

We show how certain topological properties of co-K{ä}hler manifolds derive from those of the Kähler manifolds which construct them. We go beyond Betti number results and describe the cohomology algebra structure of co-Kähler manifolds. As a consequence, we prove that co-Kähler manifolds satisfy the Toral Rank Conjectur…

The paper examines deformations of pseudoholomorphic curves in a nearly Kähler sphere.

Paper develops theory of transverse generalized complex structures and proves a key lemma.

Study on harmonic flow of Spin(7)-structures in 8D manifolds.

Study jets of flat partial connections in foliations.

No projective structure found on foliations of elliptic curves.

Study on stability of Sasaki structures under deformations.

The transversal twistor space of a foliation F of an even codimension is the bundle ZF of the complex structures of the fibers of the transversal bundle of F. On ZF, there exists a foliation F' by covering spaces of the leaves of F, and any Bott connection of F produces an ordered pair (I,J) of transversal almost compl…

We classify positive transversal torus knots in tight contact structures up to transversal isotopy.

Study simplifies classification of foliations with specific geometric structures.

Engel manifolds show transverse tori can be made to have various formal invariants.

We study the classification of special almost hermitian manifolds in Gray and Hervella's type classes. We prove that the exterior derivatives of the symplectic form and the complex volume form contain all the information about the intrinsic torsion of the $\SUn(n)$-structure. Furthermore, we apply the obtained results …

Study classifies twist knots with maximal self-linking number in S^3.

Classifies Legendrian and transverse torus knots in $S^3$.

A twistor construction of the hierarchy associated with the hyper-Kähler equations on a metric (the anti-self-dual Einstein vacuum equations, ASDVE, in four dimensions) is given. The recursion operator R is constructed and used to build an infinite-dimensional symmetry algebra and in particular higher flows for the hyp…

Study on transverse knots and their neighborhoods, proving unique standard neighborhoods and destabilization results.

Paper shows non-CSC HCMU metrics can't be isometrically immersed into 3D space forms.

We characterize the oriented Seifert-fibered three-manifolds which admit positive, transverse contact structures.

For any complete noncompact K$\ddot{a}$hler manifold with nonnegative and bounded holomorphic bisectional curvature,we provide the necessary and sufficient condition for non-ancient solution to the Ricci flow in this paper.

New theorem allows transverse links to be braided with rational book structure.