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 …

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

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

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

Given a convex body $K \subset \mathbb{R}^n$ with the barycenter at the origin we consider the corresponding K{ä}hler-Einstein equation $e^{-Φ} = \det D^2 Φ$. If $K$ is a simplex, then the Ricci tensor of the Hessian metric $D^2 Φ$ is constant and equals $\frac{n-1}{4(n+1)}$. We conjecture that the Ricci tensor of $D^2…

The paper studies Ricci curvature on Kähler-Ricci flow.

Let $M^n$ be a compact K$\ddot{a}$hler manifold with almost nonnegative Ricci curvature and nonzero first Betti number. We show that the holomorphic Euler number of $M^n$ vanishes, which gives a new obstruction for compact complex manifolds admitting K$\ddot{a}$hler metrics with almost nonnegative Ricci curvature. A cr…

In this paper, we show that any compact K$ä$hler manifold homotopic to a compact Riemannian manifold with negative sectional curvature admits a K$ä$hler-Einstein metric of general type. Moreover, we prove that, on a compact symplectic manifold $X$ homotopic to a compact Riemannian manifold with negative sectional curva…

In this paper, metric reduction in generalized geometry is investigated. We show how the Bismut connections on the quotient manifold are obtained from those on the original manifold. The result facilitates the analysis of generalized K$\ddot{a}$hler reduction, which motivates the concept of metric generalized principal…

Positive scalar curvature implies small 2-systoles in Kähler manifolds

Continuity of complex Monge-Ampère potentials on Kähler manifolds.

Anti-diagonal toric generalized K$\ddot{a}$hler structures of symplectic type on a compact toric symplectic manifold were investigated in \cite{Wang2} . In this article, we consider \emph{general} toric generalized K$\ddot{a}$hler structures of symplectic type, without requiring them to be anti-diagonal. Such a structu…

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…

We introduce a class of almost homogeneous varieties contained in the class of spherical varieties and containing horospherical varieties as well as complete symmetric varieties. We develop K{ä}hler geometry on these varieties, with applications to canonical metrics in mind, as a generalization of the Guillemin-Abreu-D…

We revisit generalized K$\ddot{a}$hler reduction introduced by Lin and Tolman in \cite{LT} from a viewpoint of geometric invariant theory. It is shown that in the strong Hamiltonian case introduced in the present paper, many well-known conclusions of ordinary K$\ddot{a}$hler reduction can be generalized without much ef…

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.

We establish a stability result for elliptic and parabolic complex Monge-Amp{è}re equations on compact K{ä}hler manifolds, which applies in particular to the K{ä}hler-Ricci flow. Dedicated to Jean-Pierre Demailly on the occasion of his 60th birthday.

Abstract: Characterizes special Kähler manifolds with specific properties.

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…

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…

We develop a parabolic pluripotential theory on compact K{ä}hler manifolds, defining and studying weak solutions to degenerate parabolic complex Monge-Amp{è}re equations. We provide a parabolic analogue of the celebrated Bedford-Taylor theory and apply it to the study of the K{ä}hler-Ricci flow on varieties with log te…

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 $…

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 …

This is a sequel of \cite{Wang}, which provides a general formalism for this paper. We mainly investigate thoroughly a subclass of toric generalized K$\ddot{a}$hler manifolds of symplectic type introduced by Boulanger in \cite{Bou}. We find torus actions on such manifolds are all \emph{strong Hamiltonian} in the sense …

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

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

We show that the bordism group of closed 3-manifolds with positive scalar curvature (psc) metrics is trivial by explicit methods. Our constructions are derived from scalar-flat K{ä}hler ALE surfaces discovered by Lock-Viaclovsky. Next, we study psc 4-manifolds with metric singularities along points and embedded circles…

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…

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.

Blowups of Kähler manifolds can inherit extremal metrics.

New extremal metrics found on Kähler manifolds.

Paper explores new Kähler metrics from old, aiming to solve YTD conjecture.

Extremal metrics exist if uniformly $K$-stable over models.

The study finds no extremal metrics for eigenvalues on compact manifolds but constructs examples for annuli.

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 Kähler metrics generalize Calabi's and relate to Fano manifolds.

Uniqueness of weighted extremal metrics on Kähler manifolds proven.

The paper classifies rotationally symmetric extremal Kähler metrics on complex manifolds.

We give a moment map interpretation of some relatively balanced metrics. As an application, we extend a result of S. K. Donaldson on constant scalar curvature Kähler metrics to the case of extremal metrics. Namely, we show that a given extremal metric is the limit of some specific relatively balanced metrics. As a coro…

We study nearly-Kahler 6-manifolds equipped with a cohomogeneity-two Lie group action for which the principal orbits are coisotropic. If the metric is complete, then we show that this last condition is automatically satisfied, and both the acting Lie group and the principal orbits are finite quotients of $S^3 \times S^…

We provide a new proof of a result of X.X.Chen and G.Tian : for a polarized extremal Kähler manifold, an extremal metric attains the minimum of the modified K-energy. The proof uses an idea of C.Li adapted to the extremal metrics using some weighted balanced metrics.

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 $s\geq n/2+4$.

Unique extremal Kähler metric found near a divisor.

Non-convex extremal length found in surface metrics.