Sets of Approximation and Interpolation in $\cx$ for manifold-valued maps

向量值函数与微分形式多项式插值的多势理论框架

A pluripotential theoretic framework for polynomial interpolation of vector-valued functions and differential forms

本文提出修正的加权多势理论框架，引入向量值权重，将标量值函数插值的渐近结果部分推广到有限维复内积空间中的向量值函数，并应用于微分形式的多项式系数插值。

Mergelyan's and Arakelian's theorems for manifold-valued maps

In this paper we show that Mergelyan's theorem holds for maps from open Riemann surfaces to Oka manifolds. This is used to prove the analogue of Arakelian's theorem on uniform approximation of holomorphic maps from closed subsets of the plane $\mathbb C$ to any compact complex homogeneous manifold.

Commutant lifting and Nevanlinna-Pick interpolation in several variables

This paper concerns a commutant lifting theorem and a Nevanlinna-Pick type interpolation result in the setting of multipliers from vector-valued Drury-Arveson space to a large class of vector-valued reproducing kernel Hilbert spaces over the unit ball in $\mathbb{C}^n$. The special case of reproducing kernel Hilbert spaces includes all natural examples of Hilbert spaces like Hardy space, Bergman space and weighted Bergman spaces over the unit ball.

Extending holomorphic maps from Stein manifolds into affine toric varieties

A complex manifold $Y$ is said to have the interpolation property if a holomorphic map to $Y$ from a subvariety $S$ of a reduced Stein space $X$ has a holomorphic extension to $X$ if it has a continuous extension. Taking $S$ to be a contractible submanifold of $X=\mathbb{C}^n$ gives an ostensibly much weaker property called the convex interpolation property. By a deep theorem of Forstnerič, the two properties are equivalent. They (and about a dozen other nontrivially equivalent properties) define the class of Oka manifolds. This paper is the first attempt to develop Oka theory for singular tar…

Approximation and interpolation of regular maps from affine varieties to algebraic manifolds

We consider the analogue for regular maps from affine varieties to suitable algebraic manifolds of Oka theory for holomorphic maps from Stein spaces to suitable complex manifolds. The goal is to understand when the obstructions to approximation or interpolation are purely topological. We propose a definition of an algebraic Oka property, which is stronger than the analytic Oka property. We review the known examples of algebraic manifolds satisfying the algebraic Oka property and add a new class of examples: smooth nondegenerate toric varieties. On the other hand, we show that the algebraic ana…

多值Sobolev映射的微分形式拉回，以及准正则映射的多值逆的准正则性

Pull-back of differential forms by multi-valued Sobolev maps, and the quasiregularity of the multi-valued inverse of a quasiregular map

本文通过Almgren框架构造准正则映射的多值逆，并发展微分形式拉回理论，证明其准正则性及高可积性。