arXiv research

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 …

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 $…

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

The paper studies Ricci curvature on Kähler-Ricci flow.

The study shows that positive scalar curvature 4-manifolds can be desingularized.

The paper proves conditions for Kähler manifolds with negative curvature.

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

Holomorphic Euler number vanishes for certain Kähler manifolds.

Positive scalar curvature implies small 2-systoles in Kähler manifolds

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

Continuity of complex Monge-Ampère potentials on Kähler manifolds.

Develops Kähler geometry on new varieties for canonical metrics.

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

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 introduces twins in Kähler and Sasaki geometry, generalizing known concepts.

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…

Study cohomology of quaternionic foliations and orbifolds.

Abstract revisits Kähler reduction using GIT, generalizing results.

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…

Abstract: Characterizes special Kähler manifolds with specific properties.

The aim of this thesis is to construct new examples of compact orbifolds $\mathcal{O}^4(Θ)$ which admit a self dual Einstein (SDE) metric of positive scalar curvature $s>0$, with a one-dimensional group of isometries. In particular we want to prove that these examples are different from those described by Boyer, Galick…

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…

We obtain a necessary and sufficient condition of existence of a K{ä}hler-Einstein metric on a $G\times G$-equivariant Fano compactification of a complex connected reductive group $G$ in terms of the associated polytope. This condition is not equivalent to the vanishing of the Futaki invariant. The proof relies on the …

Stability proven for complex equations on Kähler manifolds.

Develops theory for Kähler-Ricci flow on singular varieties.

The paper proves stability for Einstein metrics with special twisted spinors.

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

Minimal surfaces in spheres are classified based on a Ricci-like condition.

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

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…

In 2D, a conjecture about the Ricci tensor of Kähler-Einstein metrics for convex bodies is verified.

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…

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

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.

We prove local well-posedness of the Schrödinger flow from $R^n$ into a compact K\{"a}hler manifold $N$ with initial data in $H^{s+1}(R^n, N)$ for $s\geq n/2+4$.

Positive projectively flat metrics on Hopf manifolds are locally conformally flat-Kähler.

The paper characterizes complex Finsler metrics that are projectively flat or dually flat.

Two conjectures on Ricci-flat metrics verified under specific conditions.

Classifies radial scalar flat metrics with a constant third coefficient.

Study of flat metrics on orbifolds and their moduli spaces.

We briefly review the hierarchy for the hyper-Kähler equations and define a notion of symmetry for solutions of this hierarchy. A four-dimensional hyper-Kähler metric admits a hidden symmetry if it embeds into a hierarchy with a symmetry. It is shown that a hyper-Kähler metric admits a hidden symmetry if it admits a ce…

The article classifies G2-structures with conformally flat metrics.

Study on 3-metrics and their evolutions in conformally flat spaces.

The paper explores Kähler and anti-Kähler structures on quasi-statistical manifolds.

Study nearly-Kahler 6-manifolds with cohomogeneity-two action, proving existence and properties.

Investigate AR-Finsler metrics for local dual flatness and projective flatness.

Study describes flat metric moduli spaces on 4D manifolds.