International Journal of Mathematics

The main objective of this paper is to survey some recent results on the Chern–Moser question concerning existence of umbilical points on three-dimensional CR submanifolds in ${ℂ}^2$.

In this paper, we describe how pluripotential methods can be applied to study weak solutions of the complex Monge–Ampère equation on compact Hermitian manifolds. We indicate the differences between Kähler and non-Kähler setting. The results include a priori estimates, existence and stability of solutions.

This paper derives several new results, as well as gives a partial survey of the recent development, on the value sharing for algebraic and holomorphic curves into ${\mathbb{P}}^n(ℂ)$.

Let $X$ be a connected open Riemann surface. Let $Y$ be an Oka domain in the smooth locus of an analytic subvariety of ${ℂ}^n$, $n\ge 1$, such that the convex hull of $Y$ is all of ${ℂ}^n$. Let ${\mathcal{𝒪}}_{\ast }(X,Y)$ be the space of nondegenerate holomorphic maps $X\to Y$. Take a holomorphic 1-form $𝜃$ on $X$, not identically zero, and let $\pi :{\mathcal{𝒪}}_{\ast }(X,Y)\to H^1(X,{ℂ}^n)$ send a map $g$ to the cohomology class of $g𝜃$. Our main theorem states that $\pi$ is a Serre fibration. This result subsumes the 1971 theorem of Kusunoki and Sainouchi that both the periods and the divisor of a holomorphic form on $X$ can be prescribed arbitrarily. It also subsumes two parametric h-principles in minimal surface theory proved by Forstnerič and Lárusson in 2016.

An optimal $L^2$ extension theorem for the bundle-valued holomorphic sections over Stein manifolds is proved as a geometric reformulation of a theorem of Guan and Zhou. As an application, a new sharp $L^2$ extension theorem is proved.

These notes are concerned with the $L^2$-Sobolev theory of the complex Green operator on pseudoconvex, oriented, bounded and closed Cauchy–Riemann (CR)-submanifolds of ${ℂ}^n$ of hypersurface type. This class of submanifolds generalizes that of boundaries of pseudoconvex domains. We first discuss briefly the CR-geometry of general CR-submanifolds and then specialize to this class. Next, we review the basic $L^2$-theory of the tangential CR operator and the associated complex Green operator(s) on these submanifolds. After these preparations, we discuss recent results on compactness and regularity in Sobolev spaces of the complex Green operator(s).

We discuss compactness of the $\overline{\partial }$-Neumann operator in the setting of weighted $L^2$-spaces on ${ℂ}^n.$ In addition we describe an approach to obtain the compactness estimates for the $\overline{\partial }$-Neumann operator. For this purpose we have to define appropriate weighted Sobolev spaces and prove an appropriate Rellich–Kondrachov lemma.

By using the Bergman representative coordinates, we give the necessary and sufficient condition for the degree of automorphisms of quasi-circular domains fixing the origin to be bounded by the resonance order, thus solving a conjecture of the author.

These are the notes for lectures given at the Sanya winter school in complex analysis and geometry in January 2016. In Sec. 1, we review the meaning of Ricci curvature of Kähler metrics and introduce the problem of finding Kähler–Einstein metrics. In Sec. 2, we describe the formal picture that leads to the notion of K-stability of Fano manifolds, which is an algebro-geometric criterion for the existence of a Kähler–Einstein metric, by the recent result of Chen–Donaldson–Sun. In Sec. 3, we discuss algebraic structure on Gromov–Hausdorff limits, which is a key ingredient in the proof of the Kähler–Einstein result. In Sec. 4, we give a brief survey of the more recent work on tangent cones of singular Kähler–Einstein metrics arising from Gromov–Hausdorff limits, and the connections with algebraic geometry.

We first give an exposition on holomorphic isometries from the Poincaré disk to polydisks and from the Poincaré disk to the product of the Poincaré disk with a complex unit ball. As an application, we provide an example of proper holomorphic map from the unit disk to the complex unit ball that is irrational, algebraic and holomorphic on a neighborhood of the closed unit disk. We also include some new results on holomorphic isometries.

Let $M$ be a strongly pseudoconvex compact complex Finsler manifold. We first introduce a class of $g$-nature metric $G_{a,b}$ for the slit holomorphic tangent bundle $\tilde{M}=T^{1,0}M\setminus \{0\}$ on $M$. Then, we define the complex horizontal Laplacian ${\square }_h$, and complex vertical Laplacian ${\square }_v$, and obtain a precise relationship among ${\square }_h$, ${\square }_v$ and the Hodge–Laplace operator $\square$ on $(\tilde{M},G_{a,b})$. As an application, we discuss the holomorphic Killing vector fields associated to $G_{a,b}$.