{"title":"凸域中的全态 Legendrian 曲线","authors":"Andrej Svetina","doi":"arxiv-2409.04197","DOIUrl":null,"url":null,"abstract":"We prove several results on approximation and interpolation of holomorphic\nLegendrian curves in convex domains in $\\mathbb{C}^{2n+1}$, $n \\geq 2$, with\nthe standard contact structure. Namely, we show that such a curve, defined on a\ncompact bordered Riemann surface $M$, whose image lies in the interior of a\nconvex domain $\\mathscr{D} \\subset \\mathbb{C}^{2n+1}$, may be approximated\nuniformly on compacts in the interior $\\mathrm{Int} \\, M$ by holomorphic\nLegendrian curves $\\mathrm{Int} \\, M \\to \\mathscr{D}$ such that the\napproximants are proper, complete, agree with the starting curve on a given\nfinite set in $\\mathrm{Int} \\, M$ to a given finite order, and hit a specified\ndiverging discrete set in the convex domain. We first show approximation of\nthis kind on bounded strongly convex domains and then generalise it to\narbitrary convex domains. As a consequence we show that any bordered Riemann\nsurface properly embeds into a convex domain as a complete holomorphic\nLegendrian curve under a suitable geometric condition on the boundary of the\ncodomain.","PeriodicalId":501142,"journal":{"name":"arXiv - MATH - Complex Variables","volume":"12 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Holomorphic Legendrian curves in convex domains\",\"authors\":\"Andrej Svetina\",\"doi\":\"arxiv-2409.04197\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We prove several results on approximation and interpolation of holomorphic\\nLegendrian curves in convex domains in $\\\\mathbb{C}^{2n+1}$, $n \\\\geq 2$, with\\nthe standard contact structure. Namely, we show that such a curve, defined on a\\ncompact bordered Riemann surface $M$, whose image lies in the interior of a\\nconvex domain $\\\\mathscr{D} \\\\subset \\\\mathbb{C}^{2n+1}$, may be approximated\\nuniformly on compacts in the interior $\\\\mathrm{Int} \\\\, M$ by holomorphic\\nLegendrian curves $\\\\mathrm{Int} \\\\, M \\\\to \\\\mathscr{D}$ such that the\\napproximants are proper, complete, agree with the starting curve on a given\\nfinite set in $\\\\mathrm{Int} \\\\, M$ to a given finite order, and hit a specified\\ndiverging discrete set in the convex domain. We first show approximation of\\nthis kind on bounded strongly convex domains and then generalise it to\\narbitrary convex domains. As a consequence we show that any bordered Riemann\\nsurface properly embeds into a convex domain as a complete holomorphic\\nLegendrian curve under a suitable geometric condition on the boundary of the\\ncodomain.\",\"PeriodicalId\":501142,\"journal\":{\"name\":\"arXiv - MATH - Complex Variables\",\"volume\":\"12 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-09-06\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - Complex Variables\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2409.04197\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Complex Variables","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.04197","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
We prove several results on approximation and interpolation of holomorphic
Legendrian curves in convex domains in $\mathbb{C}^{2n+1}$, $n \geq 2$, 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 $\mathscr{D} \subset \mathbb{C}^{2n+1}$, may be approximated
uniformly on compacts in the interior $\mathrm{Int} \, M$ by holomorphic
Legendrian curves $\mathrm{Int} \, M \to \mathscr{D}$ such that the
approximants are proper, complete, agree with the starting curve on a given
finite set in $\mathrm{Int} \, M$ 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 bordered Riemann
surface properly embeds into a convex domain as a complete holomorphic
Legendrian curve under a suitable geometric condition on the boundary of the
codomain.