Discs in hulls of real immersions to Stein manifolds

The homotopy principle in complex analysis: a survey

This is a survey on the homotopy principle in complex analysis on Stein manifolds, also called the Oka principle in this context. We concentrate on the following topics: the Oka-Grauert principle (classification of holomorphic vector bundles); the homotopy principle for holomorphic mappings from Stein manifolds and, more generally, for sections of holomorphic submersions with sprays; question on removability of intersections of holomorphic mappings with complex subvarieties; embeddings and immersions of Stein manifolds in affine spaces of minimal dimension; embeddings of open Riemann surfaces …

On regular Stein neighborhoods of a union of two maximal totally real subspaces in $\mathbb{C}^n$

We present a construction of regular Stein neighborhoods of a union of maximally totally real subspaces $M=(A+iI)\mathbb{R}^n$ and $N=\mathbb{R}^n$ in $\mathbb{C}^n$, provided that the entries of a real $n \times n$ matrix $A$ are sufficiently small. Our proof is based on a local construction of a suitable plurisubharmonic function $ρ$ near the origin, such that the sublevel sets of $ρ$ are strongly pseudoconvex and admit strong deformation retraction to $M\cup N$. We also give the application of this result to totally real immersions of real $n$-manifolds in $\mathbb{C}^n$ with only finitely …

Stein流形的驯顺族的新例子

New examples of tame families of Stein manifolds

本文研究偶数维开流形上Stein结构的连续族，定义驯顺性并分析保持该性质的构造，给出新的驯顺Stein族例子，其满足Oka原理。

Stein fillability and the realization of contact manifolds

There is an intrinsic notion of what it means for a contact manifold to be the smooth boundary of a Stein manifold. The same concept has another more extrinsic formulation, which is often used as a convenient working hypothesis. We give a simple proof that the two are equivalent. Moreover it is shown that, even though a border always exists, it's germ is not unique; nevertheless the germ of the Dolbeault cohomology of any border is unique. We also point out that any Stein fillable compact contact 3- manifold has a geometric realization in C^4 via an embedding, or in C^3 via an immersion.

On a Conjecture of Harvey and Lawson

We consider complex projective space P^{n} and a smooth closed curve gamma in P^{n}. Harvey and Lawson have defined the notion of the projective hull \hat{K} of a compact subset K in P^n. This concept is an analogue of the polynomial hull of compact subsets of C^{n}. In the present note we study the relation between the following two properties of the curve gamma: (1) \hat{gamma} - gamma is a one-dimensional complex analytic subvariety of P^{n} - gamma, and (2) There exists a Stein subdomain of P^{n} which contains the projective hull \hat{gamma} of gamma.

Noncritical holomorphic functions on Stein manifolds

We prove that every Stein manifold X of dimension n admits [(n+1)/2] holomorphic functions with pointwise independent differentials, and this number is maximal for every n. In particular, X admits a holomorphic function without critical points; this extends a result of Gunning and Narasimhan from 1967 who constructed such functions on open Riemann surfaces. Furthermore, every surjective complex vector bundle map from the tangent bundle TX onto the trivial bundle of rank q < n=dim X is homotopic to the differential of a holomorphic submersion of X to C^q. It follows that every complex subbundle…

Minimal surfaces in minimally convex domains

In this paper, we prove that every conformal minimal immersion of a compact bordered Riemann surface $M$ into a minimally convex domain $D\subset \mathbb{R}^3$ can be approximated, uniformly on compacts in $\mathring M=M\setminus bM$, by proper complete conformal minimal immersions $\mathring M\to D$. We also obtain a rigidity theorem for complete immersed minimal surfaces of finite total curvature contained in a minimally convex domain in $\mathbb{R}^3$, and we characterize the minimal surface hull of a compact set $K$ in $\mathbb{R}^n$ for any $n\ge 3$ by sequences of conformal minimal disks…

每个格罗莫夫椭圆域都是亚纯伪凸的

Every Gromov elliptic domain is meromorphically pseudoconvex

该研究证明每个格罗莫夫椭圆域都是亚纯伪凸的，明确了Stein流形中域的格罗莫夫椭圆性与Oka、Stein等性质的等价关系，还得出两类Oka流形非椭圆及射影空间中特定域非椭圆的结论。

Stein流形的tame族的Oka原理

The Oka principle for tame families of Stein manifolds

该研究引入Stein流形的tame族概念，证明了tame族到Oka流形映射的Oka原理、相关全纯向量丛的Oka-Weil定理及同构类的Oka原理，刻画了tameness的条件并给出非tame族反例。

Oka properties of ball complements

Let $n>1$ be an integer. We prove that holomorphic maps from Stein manifolds $X$ of dimension $<n$ to the complement $\mathbb{C}^n\setminus L$ of a compact convex set $L\subset\mathbb{C}^n$ satisfy the basic Oka property with approximation and interpolation. If $L$ is polynomially convex then the same holds when $2\dim X < n$. We also construct proper holomorphic maps, immersions and embeddings $X\to\mathbb{C}^n$ with additional control of the range, thereby extending classical results of Remmert, Bishop and Narasimhan.

部分全纯关系的局部 h-原理

Local H-Principles for Partial Holomorphic Relations

本文通过引入部分全纯关系的实化概念，证明了当开部分全纯关系的实化满足相对 h-原理时，Stein 流形上的形式解可在 Lagrangian 骨架邻域内形变为全纯解，从而将局部 h-原理推广到复偶接触、复 Engel 及复共形辛结构等情形。

Oka-1 流形

Oka-1 manifolds

本文研究了 Oka-1 流形的性质，证明其在特定条件下具有不变性，并提供了多种例子和应用。