首页/论文/2001.04379

arXiv 2001.043792020-01math.CVmath.CA

Mergelyan approximation theorem for holomorphic Legendrian curves

Franc Forstneric

备注：: Anal. PDE, to appear
发表信息：: Anal. PDE 15(4):983-1010, 2022
DOI：: 10.2140/apde.2022.15.983

英文摘要：

In this paper, we prove a Mergelyan type approximation theorem for immersed holomorphic Legendrian curves in an arbitrary complex contact manifold $(X,ξ)$. Explicitly, we show that if $S$ is a compact admissible set in a Riemann surface $M$ and $f:S\to X$ is a $ξ$-Legendrian immersion of class $\mathscr{C}^{r+2}(S,X)$ for some $r\ge 2$ which is holomorphic in the interior of $S$, then $f$ can be approximated in the $\mathscr{C}^r(S,X)$ topology by holomorphic Legendrian embeddings from open neighbourhoods of $S$ into $X$. This result has numerous applications, some of which are indicated in the paper. In particular, by using Bryant's correspondence for the Penrose twistor map $\mathbb{CP}^3\to S^4$ we show that a Mergelyan approximation theorem and the Calabi-Yau property hold for superminimal surfaces in the $4$-sphere $S^4$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

Holomorphic approximation: the legacy of Weierstrass, Runge, Oka-Weil, and Mergelyan

In this paper we survey the theory of holomorphic approximation, from the classical 19th century results of Runge and Weierstrass, continuing with the 20th century work of Oka and Weil, Mergelyan, Vitushkin and others, to the most recent ones on higher dimensional manifolds. The paper includes some new results and applications of this theory, especially to manifold-valued maps.

2020-02-14math.CV

Holomorphic Approximation via Dolbeault Cohomology

The purpose of this paper is to study holomorphic approximation and approximation of $\bar\partial$-closed forms in complex manifolds of complex dimension $n\geq 1$. We consider extensions of the classical Runge theorem and the Mergelyan property to domains in complex manifolds for the smooth and the $L^2$ topology. We characterize the Runge or Mergelyan property in terms of certain Dolbeault cohomology groups and some geometric sufficient conditions are given.

2020-01-13math.CV

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.

2018-11-16math.CV

Weighted $L^2$ version of Mergelyan and Carleman approximation

We study the density of polynomials in $H^2(E,φ)$, the space of square integrable functions with respect to $e^{-φ}dm$ and holomorphic on the interior of $E$ in $\mathbb{C}$, where $φ$ is a subharmonic function and $dm$ is a measure on $E$. We give a result where $E$ is the union of a Lipschitz graph and a Carathéodory domain, that we state as a weighted $L^2$-version of the Mergelyan theorem. We also prove a weighted $L^2$-version of the Carleman theorem. Keywords: Mergelyan theorem, Carleman theorem, Weighted $L^2$- spaces, Rectifiable non-Lipschitz arc

2020-04-17math.CV

开黎曼曲面族上的Runge和Mergelyan定理

Runge and Mergelyan theorems on families of open Riemann surfaces

Faculty of Mathematics and Physics, University of Ljubljana（卢布尔雅那大学数学与物理学院）；Institute of Mathematics, Physics, and Mechanics（数学、物理和力学研究所）

本文针对带有连续或光滑依赖于参数的Hölder类复结构族的开定向曲面，构造了到任意Oka流形的全纯映射族，并证明了Runge逼近、Mergelyan逼近和Weierstrass插值定理，应用于构造定向全纯浸入和共形极小浸入族。

2026-05-24math.CVmath.DG

Darboux charts around holomorphic Legendrian curves and applications

In this paper, we find a holomorphic Darboux chart around any immersed noncompact holomorphic Legendrian curve in a complex contact manifold $(X,ξ)$. By using such a chart, we show that every holomorphic Legendrian immersion $R\to X$ from an open Riemann surface can be approximated on relatively compact subsets by holomorphic Legendrian embeddings, and every holomorphic Legendrian immersion $M\to X$ from a compact bordered Riemann surface is a uniform limit of topological embeddings $M\hookrightarrow X$ such that $\mathring M\hookrightarrow X$ is a complete holomorphic Legendrian embedding. We…

2017-06-16math.CVmath.DG

更多相关论文收起更多论文

Holomorphic Legendrian curves

In this paper we study holomorphic Legendrian curves in the standard holomorphic contact structure on $\mathbb{C}^{2n+1}$ for any $n\in\mathbb{N}$. We provide several approximation and desingularization results which enable us to prove general existence theorems, settling some of the open problems in the subject. In particular, we show that every open Riemann surface $M$ admits a proper holomorphic Legendrian embedding $M\hookrightarrow\mathbb{C}^{2n+1}$, and we prove that for every compact bordered Riemann surface $M=\mathring M\cup bM$ there exists a topological embedding $M\hookrightarrow \…

2017-06-05math.CVmath.DG

全纯勒让德曲线的逼近及其射流插值

Approximation of holomorphic Legendrian curves with jet-interpolation

该文证明全纯勒让德曲线的插值与逼近定理，包括可数点集位于浸入曲面及Carleman型逼近，并给出适当嵌入条件。

2023-05-16math.CV

Proper holomorphic Legendrian curves in $SL_2(\mathbb{C})$

In this paper we prove that every open Riemann surface properly embeds in the Special Linear group $SL_2(\mathbb{C})$ as a holomorphic Legendrian curve, where $SL_2(\mathbb{C})$ is endowed with its standard contact structure. As a consequence, we derive the existence of proper, weakly complete, flat fronts in the real hyperbolic space $\mathbb{H}^3$ with arbitrary complex structure.

2016-11-02math.CVmath.DG

Holomorphic Legendrian curves in projectivised cotangent bundles

We study holomorphic Legendrian curves in the standard complex contact structure on the projectivised cotangent bundle $X=\mathbb P(T^*Z)$ of a complex manifold $Z$ of dimension at least $2$. We provide a detailed analysis of Legendrian curves degenerating to vertical curves and obtain several approximation and general position theorems. In particular, we prove that any vertical holomorphic curve $M\to X$ from a compact bordered Riemann surface $M$ can be deformed to a horizontal Legendrian curve by an arbitrarily small deformation. A similar result is proved in the parametric setting, provide…

2022-02-06math.CVmath.SG

Holomorphic Legendrian curves in $\mathbb{CP}^3$ and superminimal surfaces in $\mathbb S^4$

We obtain a Runge approximation theorem for holomorphic Legendrian curves and immersions in the complex projective $3$-space $\mathbb{CP}^3$, both from open and compact Riemann surfaces, and we prove that the space of Legendrian immersions from an open Riemann surface into $\mathbb{CP}^3$ is path connected. We also show that holomorphic Legendrian immersions from Riemann surfaces of finite genus and at most countably many ends, none of which are point ends, satisfy the Calabi-Yau property. Coupled with the Runge approximation theorem, we infer that every open Riemann surface embeds into $\math…

2022-02-06math.CVmath.DG

凸域中的全纯勒让德曲线

Holomorphic Legendrian curves in convex domains

本文研究高维复空间凸域中的全纯勒让德曲线，证明带边黎曼曲面上的曲线可由真、完备且满足有限阶插值与命中发散离散集条件的全纯勒让德曲线逼近，并推出相应真嵌入结论。

2024-09-06math.CV

The Oka principle for holomorphic Legendrian curves in $\mathbb C^{2n+1}$

Let $M$ be a connected open Riemann surface. We prove that the space $\mathscr L(M,\mathbb C^{2n+1})$ of all holomorphic Legendrian immersions of $M$ into $\mathbb C^{2n+1}$, $n\geq 1$, endowed with the standard holomorphic contact structure, is weakly homotopy equivalent to the space $\mathscr C(M,\mathbb S^{4n-1})$ of continuous maps from $M$ to the sphere $\mathbb S^{4n-1}$. If $M$ has finite topological type, then these spaces are homotopy equivalent. We determine the homotopy groups of $\mathscr L(M,\mathbb C^{2n+1})$ in terms of the homotopy groups of $\mathbb S^{4n-1}$. It follows that …

2017-05-15math.CVmath.DG

Approximation on compact sets of functions and all derivatives

In Mergelyan type approximation we uniformly approximate functions on compact sets K by polynomials or rational functions or holomorphic functions on varying open sets containing K. In the present paper we consider analogous approximation, where uniform convergence on K is replaced by uniform approximation on K of all order derivatives.

2020-06-03math.CAmath.CVmath.FA

C^n 中加权全纯 Besov 空间的逼近

Approximation in weighted holomorphic Besov spaces in C^n

本文研究多圆柱和单位球中全纯函数的加权 Bergman 与 Besov 空间，寻求非径向权函数的 Mergelyan 型条件以保证函数伸缩按范数收敛且解析多项式在空间中稠密。

2023-04-02math.CVmath.FA

Distributions, first integrals and Legendrian foliations

We study germs of holomorphic distributions with "separated variables'. In codimension one, a well know example of this kind of distribution is given by the canonical contact structure on $\mathbb{P}^{2m+1}$ . Another example is the Darboux distribution, which gives the normal local form of any contact structure. Given a germ $D$ of holomorphic distribution with separated variables in $(\mathbb{C}^n,0)$, we show that there exists , for some $κ\in \mathbb{Z}_{\geq 0}$ related to the Taylor coefficients of $D$, a holomorphic submersion $H_{D}: (\mathbb{C}^n,0) \rightarrow (\mathbb{C}^κ,0)$ such …

2022-05-18math.AGmath.CV

特殊线性群中的全纯零曲线

Holomorphic null curves in the special linear group

本文建立 ${\rm SL}_2(\mathbb{C})$ 中全纯零浸入的多种逼近与插值定理，并证明任意开黎曼曲面均可真全纯零嵌入该群，从而解决相关公开问题。

2025-07-25math.CVmath.DG

开黎曼曲面到黎曼球面的浸入

Immersions of open Riemann surfaces into the Riemann sphere

该研究证明开黎曼曲面到黎曼球面的全纯浸入空间与对应连续映射空间弱同伦等价，给出其道路连通分支数，并证明用于主定理的参数化Mergelyan逼近定理。

2021-04-22math.CV

具有康托末端的极小曲面的 Calabi-Yau 问题

The Calabi-Yau problem for minimal surfaces with Cantor ends

本文证明任意连通紧致或带边黎曼面都含有一个康托集，其补集可完备共形极小浸入到有界区域中的 $\mathbb R^3$，并将类似结论推广到多种全纯、勒让德及超极小浸入情形。

2022-07-14math.CVmath.DG

Oka-1 流形

Oka-1 manifolds

Universidad de Granada（格拉纳达大学）；University of Ljubljana（卢布尔雅那大学）；Institute of Mathematics, Physics, and Mechanics（数学、物理与力学研究所）

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。

2025-04-10math.CV

Complete minimal surfaces densely lying in arbitrary domains of $\mathbb{R}^n$

In this paper we prove that, given an open Riemann surface $M$ and an integer $n\ge 3$, the set of complete conformal minimal immersions $M\to\mathbb{R}^n$ with $\overline{X(M)}=\mathbb{R}^n$ forms a dense subset in the space of all conformal minimal immersions $M\to\mathbb{R}^n$ endowed with the compact-open topology. Moreover, we show that every domain in $\mathbb{R}^n$ contains complete minimal surfaces which are dense on it and have arbitrary orientable topology (possibly infinite); we also provide such surfaces whose complex structure is any given bordered Riemann surface. Our method of p…

2016-11-15math.CVmath.DG

Null curves and directed immersions of open Riemann surfaces

In this paper we study holomorphic immersions of open Riemann surfaces into C^n whose derivative lies in a conical algebraic subvariety A of C^n that is smooth away from the origin. Classical examples of such A-immersions include null curves in C^3 which are closely related to minimal surfaces in R^3, and null curves in SL_2(C) that are related to Bryant surfaces. We establish a basic structure theorem for the set of all A-immersions of a bordered Riemann surface, and we prove several approximation and desingularization theorems. Assuming that A is irreducible and is not contained in any hyper…

2013-06-24math.AGmath.CVmath.DG