Holomorphic Legendrian curves in convex domains

Svetina, Andrej (author)

We prove several results on approximation and interpolation of holomorphic Legendrian curves in convex domains in ${\mathbb C}^{2n+1}$, $n \ge 2$, with the standard contact structure. Namely, we show that such a curve, defined on a compact bordered Riemann surface $M$, whose image lies in the interior of a convex domain ${\mathcal D} \subset {\mathbb C}^{2n+1}$, may be approximated uniformly on compacts in the interior ${\rm Int} M$ by holomorphic Legendrian curves ${\rm Int} M \to {\mathcal D}$ such that the approximants are proper, complete, agree with the starting curve on a given finite set in ${\rm Int} M$ to a given finite order, and hit a specified diverging discrete set in the convex domain. We first show approximation of this kind on bounded strongly convex domains and then generalise it to arbitrary convex domains. As a consequence we show that any compact bordered Riemann surface properly embeds into a convex domain as a complete curve under a suitable geometric condition on the boundary of the codomain.

• Language: English
• Keywords: holomorphic Legendrian curve, convex domain, complete Legendrian embedding
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Year: 2025
• Number of pages: 32 str.
• Numbering: Vol. 35, iss. 1, art. 37
• PID: 20.500.12556/RUL-175141
• UDC: 517.5
• ISSN on article: 1050-6926
• DOI: 10.1007/s12220-024-01872-7
• COBISS.SI-ID: 218411523
• Publication date in RUL: 17.10.2025

Record is a part of a journal

• Title: The Journal of geometric analysis
• Shortened title: J. geom. anal.
• Publisher: Springer, Mathematica Josephina
• ISSN: 1050-6926
• COBISS.SI-ID: 30685696

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Projects

• Funder: ARRS - Slovenian Research Agency
• Funding programme: Young researchers

• Funder: ARRS - Slovenian Research Agency
• Project number: P1-0291
• Name: Analiza in geometrija