arXiv research

Study shows Bergman metric is non-Einstein for certain domains.

This paper is a sequel to \cite{Choi} in Math. Ann. In that paper we studied the subharmonicity of Kähler-Einstein metrics on strongly pseudoconvex domains of dimension greater than or equal to $3$. In this paper, we study the variations Kähler-Einstein metrics on bounded strongly pseudoconvex domains of dimension $2$.…

Investigate pseudoconvexity of locally trivial holomorphic ball bundles over compact Riemann surfaces.

Smooth families of biholomorphisms between strongly pseudoconvex domains are shown to be smooth.

The Bergman-Szegő kernel is analyzed for weakly pseudoconvex CR manifolds of finite type.

We obtain an embedding theorem for compact strongly pseudoconvex CR manifolds which are bounadries of some complete Hermitian manifolds. We use this to compactify some negatively curved Kaehler manifolds with compact strongly pseudoconvex boundary. An embedding theorem for Sasakian manifolds is also derived.

The pseudoconvex and disprisoning conditions for geodesics of linear connections are extended to the solution curves of general homogeneous sprays. The main result is that pseudoconvexity and disprisonment are jointly stable in the fine topology on the space of all homogeneous sprays of any degree of homogeneity.

Study on eigenvalues of complex Hessian operator on pseudoconvex manifolds.

The paper proves a conjecture about the Bergman metric of real analytic domains.

Compactifies CR structures for complex hyperbolic manifolds.

Study intrinsic volume forms on complex hypersurfaces.

The paper proves a disk's energy minimizer is holomorphic and calculates its Morse index.

Paper introduces a new Poisson kernel for strongly pseudoconvex domains.

In this note we shall prove that the complete Kähler-Einstein volume form on a bounded strongly pseudoconvex domain with $C^{\infty}$-boundary is the normalized limit of a sequence of Bergman kernels.

Modelled on a real hypersurface in a quaternionic manifold, we introduce a quaternionic analogue of CR structure, called quaternionic CR structure. We define the strong pseudoconvexity of this structure as well as the notion of quaternionic pseudohermitian structure. Following the construction of the Tanaka-Webster con…

For any pseudoconvex Runge domain $Ω\subset\mathbb{C}^2$ we prove that every closed discrete subset in $Ω$ is contained in a properly embedded complex curve in $Ω$ with any prescribed topology (possibly infinite).

Study CR Yamabe constant and CR structures on manifolds.

We prove a local boundary regularity result for the complete Kahler-Einstein metrics of negative Ricci curvature near strictly pseudoconvex boundary point. We also study the asymptotic behaviour of their holomorphic bisectional curvatures near such points.

Study Kähler-Ricci solitons on bounded domains, proving they are Kähler-Einstein.

We study Kahler manifolds-with-boundary, not necessarily compact, with weakly pseudoconvex boundary, each component of which is compact. If such a manifold $K$ has $l\ge2$ boundary components (possibly $l=\infty$), then it has first betti number at least $l-1$, and the Levi form of any boundary component is zero. If $K…

This short paper gives a constraint on Chern classes of closed strictly pseudoconvex CR manifolds (or equivalently, closed holomorphically fillable contact manifolds) of dimension at least five. We also see that our result is ''optimal'' through some examples.

We refine estimates introduced by Balogh and Bonk, to show that the boundary extensions of isometries between smooth strongly pseudoconvex domains in $\C^n$ are conformal with respect to the sub-Riemannian metric induced by the Levi form. As a corollary we obtain an alternative proof of a result of Fefferman on smooth …

The paper proves that a fiberwise Kähler-Ricci flow is positive for all time on a family of bounded strongly pseudoconvex domains.

Two-dimensional domains with Kähler-Einstein Bergman metrics are biholomorphic to the unit ball.

In this paper, we obtain optimal $L^2$ extension of holomorphic sections of a holomorphic vector bundle from subvarieties in weakly pseudoconvex Kähler manifolds. Moreover, in the case of line bundle the Hermitian metric is allowed to be singular.

We construct a complete proper holomorphic embedding from any strictly pseudoconvex domain with $\mathcal{C}^2$-boundary in $\mathbb{C}^n$ into the unit ball of $\mathbb{C}^N$, for $N$ large enough, thereby answering a question of Alarcon and Forstneric.

We establish the existence of Kähler-Ricci flow on pseudoconvex domains with general initial metric without curvature bounds. Moreover we prove that this flow is simultaneously complete, and its normalized version converge to the complete Kähler-Einstein metric, which generalizes Topping's works on surfaces.

We consider a class of complete Kahler manifolds with a strictly pseudoconvex boundary at infinity. After studying its asymptotic geometry, we formulate a conjecture in the Kahler-Einstein case relating the bottom of spectrum to the CR geometry on the boundary. We prove some partial results.

We introduce analogues of a map due to Rossi and show how they can be used to explicitly determine all covers of certain homogeneous strongly pseudoconvex 3-dimensional hypersurfaces that appear in the classification obtained by E. Cartan in 1932.

New method constructs potential functions for Kähler-Einstein metrics.

Let $p:X\rightarrow Y$ be a surjective holomorphic mapping between Kähler manifolds. Let $D$ be a bounded smooth domain in $X$ such that every generic fiber $D_y:=D\cap p^{-1}(y)$ for $y\in Y$ is a strongly pseudoconvex domain in $X_y:=p^{-1}(y)$, which admits the complete Kähler-Einstein metric. This family of Kähler-…

In this note, we affirm the partial answer to the long open Conjecture which states that any closed embeddable strictly pseudoconvex CR $3$-manifold admits a contact form $θ$ with the vanishing CR $Q$-curvature. More precisely, we deform the contact form according to an CR analogue of $Q$%-curvature flow in a closed st…

Study curvature in holomorphic fibration fields.

Study Szegő kernel on non-compact CR manifolds with specific conditions.

Establishes a lower bound for Kähler hyperbolicity modulus in hyperconvex domains and bounded strongly pseudoconvex domains.

Solves embedding problem for 5D manifolds into Calabi-Yau 3-folds.

The paper explores properties of CR hypersurfaces and their flatness.

On a bounded strictly pseudoconvex domain in $\mathbb{C}^n$, $n>1$, the smoothness of the Cheng-Yau solution to Fefferman's complex Monge-Ampere equation up to the boundary is obstructed by a local curvature invariant of the boundary. For bounded strictly pseudoconvex domains in $\mathbb{C}^2$ which are diffeomorphic t…

The first result is the semicontinuity of automorphism groups for the collection of complex two-dimensional bounded pseudoconvex domains with smooth boundary of finite D'Angelo type. The method of proof is new so that it simplifies the previous proof of earlier semicontinuity theorems on bounded strongly pseudoconvex d…

The paper studies the curvature behavior near the boundary of certain domains.

The $Q$-prime curvature is a local invariant of pseudo-Einstein contact forms on integrable strictly pseudoconvex CR manifolds. The transformation law of the $Q$-prime curvature under scaling is given in terms of a differential operator, called the $P$-prime operator, acting on the space of CR pluriharmonic functions. …

The metrics of S. Y. Cheng and S.-T. Yau are considered on a strictly pseudoconvex domains in a complex manifold. Such a manifold carries a complete Kähler-Einstein metric if and only if its canonical bundle is positive. We consider the restricted case in which the CR structure on $\partial M$ is normal. In this case M…

The study shows algebraic Bergman kernels imply finite type boundaries in complex domains.

The paper proves a Schwarz lemma for weakly Kähler-Finsler manifolds.

For a smooth strictly pseudoconvex hypersurface in a complex manifold, we give a necessary and sufficient condition for being CR-diffeomorphic to a real-analytic CR manifold. Our condition amounts to a holomorphic extension property for the canonically associated function expressing $2$-jets of the formal Segre varieti…

Study of pseudoconvex 3-manifolds in complex surfaces.

Study solves complex equation on specific types of manifolds.

We establish inequalities for the eigenvalues of the sub-Laplace operator associated with a pseudo-Hermitian structure on a strictly pseudoconvex CR manifold. Our inequalities extend those obtained by Niu and Zhang \cite{NiuZhang} for the Dirichlet eigenvalues of the sub-Laplacian on a bounded domain in the Heisenberg …