arXiv research

Proves conformal equivalence of visual metrics in smooth pseudoconvex domains.

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

Study relationships between submanifolds and ambient Kahler 4-manifolds' fundamental groups.

New index connects Diederich-Fornæss index to boundary conditions.

We derive a sufficient condition on a bounded pseudoconvex domain $Ω\subset\mathbb{C}^2$ with smooth boundary such that $-(-ρ)^η$ is plurisubharmonic on $Ω$ for $η>0$ arbitrarily close to $1$ (the supremum of $η$ is called Diederich-Fornæss index, see Definition (df)). This condition (see Theorem prop) extends a theore…

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…

Paper connects CR geometry conditions to closed range of $\bar\partial$-operator.

In this paper, we consider real hypersurfaces $M$ in $\Bbb C^3$ (or more generally, 5-dimensional CR manifolds of hypersurface type) at uniformly Levi degenerate points, i.e. Levi degenerate points such that the rank of the Levi form is constant in a neighborhood. We also require the hypersurface to satisfy a certain s…

Let $Ω$ be a pseudoconvex domain with $C^2$-smooth boundary in $\mathbb CP^n$. We prove that the $\bar\partial-Neumann operator$N$exists for$(p,q)$-forms on$Ω$. Furthermore, there exists a$t_0>0$such that the operators$N$,$\bar\partial^*N$,$\bar\partial N$ and the Bergman projection are regular in the Sobolev …

We address the problem of existence and uniqueness of a Levi-flat hypersurface $M$ in $C^n$ with prescribed compact boundary $S$ for $n\ge3$. The situation for $n\ge3$ differs sharply from the well studied case $n=2$. We first establish necessary conditions on $S$ at both complex and CR points, needed for the existence…

Develops new approach to recover CR structures from their Levi foliations.

For a subRiemannian manifold and a given Riemannian extension of the metric, we define a canonical global connection. This connection coincides with both the Levi-Civita connection on Riemannian manifolds and the Tanaka-Webster connection on strictly pseudoconvex CR manifolds. We define a notion of normality generalizi…

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

Study shows nonvanishing CR curvature on Grauert tube boundaries.

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 $2$.…

Flow on pseudoconvex domains without curvature bounds.

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.

Holomorphic family of strongly pseudoconvex domains in Kähler manifolds are studied.

Every discrete subset in a complex domain is in a complex curve.

The paper sets a limit on CR manifold properties.

Proves local boundary regularity for negative curvature Kahler-Einstein metrics.

Optimal $L^2$ extension of sections from subvarieties in Kähler manifolds.

Study intrinsic volume forms on complex hypersurfaces.

CR-harmonic maps defined for pseudoconvex manifolds.

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

Study on geometric properties of plurisubharmonic functions in strongly pseudoconvex domains.

Study on symmetric domains with Bergman metric properties.

Paper introduces a new Poisson kernel for strongly pseudoconvex domains.

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…

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.

Study CR Yamabe constant and CR structures on manifolds.

Holomorphic motions can't map to complex domains.

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

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.

Researchers study curvature invariants on complex domains, finding rigidity for unit balls.

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 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.

New method constructs potential functions for Kähler-Einstein metrics.

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.

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