arXiv research

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.

Study on eigenvalues of complex Hessian operator on pseudoconvex manifolds.

Compactifies CR structures for complex hyperbolic manifolds.

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

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

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

CR-harmonic maps defined for pseudoconvex manifolds.

Study CR Yamabe constant and CR structures on manifolds.

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.

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

Study intrinsic volume forms on complex hypersurfaces.

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…

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

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

Embeds CR manifolds into complex spaces using equivariant actions.

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

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.

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…

CR Q-curvature flow solves CR manifold curvature conjecture.

Study curvature in holomorphic fibration fields.

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. …

Study of pseudoconvex 3-manifolds in complex surfaces.

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

Study solves complex equation on specific types of manifolds.

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…

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

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

We solve $\square_b$ on a class of non-compact 3-dimensional strongly pseudoconvex CR manifolds via a certain conformal equivalence. The idea is to make use of a related $\square_b$ operator on a compact 3-dimensional strongly pseudoconvex CR manifold, which we solve using a pseudodifferential calculus. The way we solv…

A complex filling of a CR manifold is said to be equivariant with respect to a CR action if the action extends to a smooth action by biholomorphisms on the whole filling. Under a noncompactness condition for the action, we describe all equivariant fillings of strongly pseudoconvex CR manifolds of dimension 3. Since the…

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

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

A locally conformally Kahler (LCK) manifold is a complex manifold admitting a Kahler covering M, such that its monodromy acts on this covering by homotheties. A compact LCK manifold is called LCK with potential if M admits an authomorphic Kahler potential. It is known that in this case it is an algebraic cone, that is,…

We study the pseudohermitian sectional curvature of a CR manifold.

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 …

Formula for Toeplitz operator kernel on CR manifolds.

The paper provides conditions for smooth CR-manifolds to be CR-diffeomorphic to real-analytic ones.

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…

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

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

Developing deformation theory for Calabi-Yau 3-folds with boundary.

We obtain a Bochner type formula and an estimate from below on the spectrum of the sublaplacian of a compact strictly pseudoconvex CR manifold.

New method constructs Stein surfaces using topological isotopy.

Study on contact forms with constant curvature on CR manifolds.

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 $2$.…

The CR Frankel conjecture is proven for spherical CR manifolds.

Let $X$ be a compact connected strongly pseudoconvex CR manifold of dimension $2n+1, n \ge 1$ with a transversal CR $S^1$-action on $X$. We introduce the Fourier components of the Ray-Singer analytic torsion on $X$ with respect to the $S^1$-action. We establish an asymptotic formula for the Fourier components of the an…

Establishes a lower bound for Kähler-Einstein distance on certain 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…