关于Lempert函数的乘积性质

On the present state of the Andersen-Lempert theory

In this survey of the Andersen-Lempert theory we present the state of the art in the study of the density property (which means that the Lie algebra generated by completely integrable holomorphic vector fields on a given Stein manifold is dense in the space of all holomorphic vector fields). There are also two new results in the paper one of which is the theorem stating that the product of Stein manifolds with the volume density property possesses such a property as well. The second one is a meaningful example of an algebraic surface without the algebraic density property. The proof of the las…

Lempert函数的准三角不等式

Quasi Triangle Inequality for the Lempert function

研究证明在具有C^1光滑边界的伪凸域上，Lempert函数满足准三角不等式，并推导了与边界正则性相关的局部界。

On a convex level set of a plurisubharmonic function and the support of the Monge-Ampère current

In this paper, we study a geometric property of a continuous plurisubharmonic function which is a solution of the Monge-Ampère equation and has a convex level set. To prove our main theorem, we show a minimum principle of a maximal plurisubharmonic function. By using our results and Lempert's results, we show a relation between the supports of the Monge-Ampère currents and complex $k$-extreme points of closed balls for the Kobayashi distance in a bounded convex domain in $\mathbb{C}^{n}$.

Manifolds with infinite dimensional group of holomorphic automorphisms and the Linearization Problem

We overview a number of precise notions for a holomorphic automorphism group to be big together with their implications, in particular we give an exposition of the notions of flexibility and of density property. These studies have their origin in the famous result of Andersén and Lempert from 1992 proving that the overshears generate a dense subgroup in the holomorphic automorphism group of $\C^n, n\ge 2$. There are many applications to natural geometric questions in complex geometry, several of which we mention here. Also the Linearization Problem, well known since the 1950 s and considered b…

关于复格林函数的乘积性质

On the product property of the pluricomplex Green function

证明了复格林函数对任意整环$D_1\subset\Bbb C^n$和$D_2\subset\Bbb C^m$都具有乘积性质$g_{D_1\times D_2}=\max\{ g_{D_1},g_{D_2}\}$.

On a boundary property of Blaschke products

A Blaschke product has no radial limits on a subset $E$ of the unit circle $T$ but has unrestricted limit at each point of $T \setminus E$ if and only if $E$ is a closed set of measure zero.