arXiv research

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.

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…

Local rigidity results for Bergman and Kähler Carathéodory metrics on 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…

Proves Hölder continuity of complex Monge-Ampère solutions.

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 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…

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

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

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 …

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

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

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

It is shown that the Ramadanov conjecture implies the Cheng conjecture. In particular it follows that the Cheng conjecture holds in dimension two.

The paper studies Fubini-Study metrics and zero distributions on CR manifolds.

Study on when Bergman metrics of domains are induced by balls.

Compactifies CR structures for complex hyperbolic manifolds.

Study on contact forms with constant curvature on CR manifolds.

We establish a lower estimate for the Kobayashi-Royden infinitesimal pseudometric on an almost complex manifold $(M,J)$ admitting a bounded strictly plurisubharmonic function. We apply this result to study the boundary behaviour of the metric on a strictly pseudoconvex domain in $M$ and to give a sufficient condition f…

Local vanishing theorems for complex spaces with smooth boundaries.

An n-dimensional strictly pseudoconvex Hartogs domain D_F can be equipped with a natural Kaehler metric g_F. In this paper we prove that if m_0g_F is balanced for a given positive integer m_0 then m_0>n and (D_F, g_F) is holomorphically isometric to an open subset of the n-dimensional complex hyperbolic space.

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.

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.

This is a survey on the correspondence between asymptotically complex hyperbolic Einstein metrics and CR structures on the boundary at infinity, which is the complex version of that between Poincaré-Einstein metrics and conformal structures. We mainly discuss existence theorems, and propose several open problems.

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

We give a necessary and sufficient condition for the smooth extension of a diffeomorphism between smooth strictly pseudoconvex domains in four real dimensional almost complex manifolds. The proof is mainly based on a reflection principle for pseudoholomorphic discs, on precise estimates of the Kobayashi-Royden infinite…

This note establishes smooth approximation from above for J-plurisubharmonic functions on an almost complex manifold (X,J). The following theorem is proved. Suppose X is J-pseudoconvex, i.e., X admits a smooth strictly J-plurisubharmonic exhaustion function. Let u be an (upper semi-continuous) J-plurisubharmonic functi…

For a strictly pseudoconvex domain in a complex manifold we define a renormalized volume with respect to the approximately Einstein complete Kähler metric of Fefferman. We compute the conformal anomaly in complex dimension two and apply the result to derive a renormalized Chern--Gauss--Bonnet formula. Relations between…

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 $2$.…

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…

In this paper, we discuss a rigidity property for holomorphic disks in Teichmüller space. In fact, we give a refinement of Tanigawa's rigidity theorem. We will also treat the rigidity property of holomorphic disks for complex manifolds. We observe the rigidity property is valid for bounded strictly pseudoconvex domains…

Paper proves a quantitative estimate for transforming almost complex structures into standard ones.

We derive variational formulas for the total Q-prime curvature under the deformation of strictly pseudoconvex domains in a complex manifold. We also show that the total Q-prime curvature agrees with the renormalized volume of such domains with respect to the complete Einstein-Kähler metric. In the appendix, by Rod Gove…

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

Any strictly pseudoconvex domain in C2 carries a complete Kahler-Einstein metric, the Cheng-Yau metric, with ``conformal infinity'' the CR structure of the boundary. It is well known that not all CR structures on the 3-sphere arise in this way. In this paper, we study CR structures on the 3-sphere satisfying a differen…

We first obtain the interior $C^{1,1}$-regularity and solvability for the degenerate real Monge-Ampère equation in a bounded, $C^3$-smooth and strictly convex domain in $\mathbb R^d$ ($d\ge2$), assuming that the boundary data is only globally $C^{1,1}$, and the $d$-th root of the nonnegative right-hand side is globally…

We study pseudo Yang-Mills fields on a compact strictly pseudoconvex CR manifold.

The scattering operators associated to an ACHE metric of Bergman type on a strictly pseudovonvex domain are a one-parameter family of CR-conformally invariant pseudodifferntial operators of Heisenberg class with respect to the induced CR structure on the boundary. In this paper, we mainly show that if the boundary Webs…

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

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 study the pseudohermitian sectional curvature of a CR manifold.

Formula for Toeplitz operator kernel on CR manifolds.

Let $M^{2n-1}$ be the smooth boundary of a bounded strongly pseudo-convex domain $Ω$ in a complete Stein manifold $V^{2n}$. Then (1) For $n \ge 3$, $M^{2n-1}$ admits a pseudo-Eistein metric; (2) For $n \ge 2$, $M^{2n-1}$ admits a Fefferman metric of zero CR Q-curvature; and (3) for a compact strictly pseudoconvex CR em…

In this note, we first give a criterion of pseudo-Einstein contact forms and then affirm the CR analogue of Frankel conjecture in a closed, spherical, strictly pseudoconvex CR manifold of nonnegative pseudohermitian curvature on the space of smooth representatives of the first Kohn-Rossi cohomology group. Moreover, we …

Paper solves complex equations on noncompact manifolds.

We build a variational theory of geodesics of the Tanaka-Webster connection on a strictly pseudoconvex CR manifold.