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 $2$.…

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

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

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

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

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

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

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.

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).

New method constructs potential functions for Kähler-Einstein metrics.

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.

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.

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-…

Local rigidity results for Bergman and Kähler Carathéodory metrics on domains.

Study curvature in holomorphic fibration fields.

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

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…

In this paper we establish a gap theorem for the complex geometry of smoothly bounded convex domains which informally says that if the complex geometry near the boundary is close to the complex geometry of the unit ball, then the domain must be strongly pseudoconvex. One consequence of our general result is the followi…

Establishes a lower bound for Kähler-Einstein distance on certain domains.

The Wong-Rosay theorem characterizes the strongly pseudoconvex domains of $\mathbb{C}^n$ by their automorphism groups. It has a lot of generalizations to other kinds of domains (for example, the weakly pseudoconvex domains). However, most of them are for domains of $\mathbb{C}^n$. In this note, we generalize the Wong-R…

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

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…

This paper describes a method to construct standard 4-balls from homotopy 4-balls in $\mathbb{C}^2$.

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

The study proves the existence of complete Kähler metrics with negative holomorphic bisectional curvature in specific domains.

The paper introduces new metrics on complex domains with specific geometric properties.

Local vanishing theorems for complex spaces with smooth boundaries.

The paper extends Newlander-Nirenberg theorem to domains with $C^2$ boundary.

Proves Hölder continuity of complex Monge-Ampère solutions.

We study how the existence of a negatively pinched Kähler metric on a domain in complex Euclidean space restricts the geometry of its boundary. In particular, we show that if a convex domain admits a complete Kähler metric, with pinched negative holomorphic bisectional curvature outside a compact set, then the boundary…

We construct new complete Einstein metrics on smoothly bounded strictly pseudoconvex domains in Stein manifolds. This is done by deforming the Kähler-Einstein metric of Cheng and Yau, the approach that generalizes the works of Roth and Biquard on the deformations of the complex hyperbolic metric on the unit ball. Recas…

New method detects non-product domains using squeezing function.

The paper studies complex Finsler metrics and their equivalence to the Kobayashi metric.

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…

We study the complete Kahler-Einstein metric in tube domains. We obtain estimates of this metric and its holomorphic bisectional curvatures near the weakly pseudoconvex boundary points.

In this paper we introduce a new class of domains in complex Euclidean space, called Goldilocks domains, and study their complex geometry. These domains are defined in terms of a lower bound on how fast the Kobayashi metric grows and an upper bound on how fast the Kobayashi distance grows as one approaches the boundary…

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. …

We study bounded pseudoconvex domains in complex Euclidean space. We define an index associated to the boundary and show this new index is equivalent to the Diederich-Fornæss index defined in 1977. This connects the Diederich-Fornæss index to boundary conditions and refines the Levi pseudoconvexity. We also prove the $…

We shall give a definition of the curvature operator for a family of weighted Bergman spaces $\{\mathcal H_t\}$ associated to a smooth family of smoothly bounded strongly pseudoconvex domains $\{D_t\}$. In order to study the boundary term in the curvature operator, we shall introduce the notion of geodesic curvature fo…

We show that the graph of a holomorphic motion of the unit disc cannot be biholomorphic to a strongly pseudoconvex domain in C n .

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 …

Introduces Levi core for CR manifolds, linking it to global invariants.

The paper proves conditions for Kähler-Einstein metrics to remain Kähler-Einstein under cscK perturbations.

An $n$-dimensional Hartogs domain $D_F$ with strongly pseudoconvex boundary can be equipped with a natural \K metric $g_F$. In this paper we prove that if $g_F$ is an extremal \K metric then $(D_F, g_F)$ is biholomorphically isometric to the $n$-dimensional complex hyperbolic space.