Every nonflat conformal minimal surface is homotopic to a proper one
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918388584939520 |
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| author | Vrhovnik, Tjasa |
| author_facet | Vrhovnik, Tjasa |
| contents | Given an open Riemann surface $M$, we prove that every nonflat conformal minimal immersion $M\to\mathbb{R}^n$ ($n\geq 3$) is homotopic through nonflat conformal minimal immersions $M\to\mathbb{R}^n$ to a proper one. If $n\geq 5$, it may be chosen in addition injective, hence a proper conformal minimal embedding. Prescribing its flux, as a consequence, every nonflat conformal minimal immersion $M\to\mathbb{R}^n$ is homotopic to the real part of a proper holomorphic null embedding $M\to\mathbb{C}^n$. We also obtain a result for a more general family of holomorphic immersions from an open Riemann surface into $\mathbb{C}^n$ directed by Oka cones in $\mathbb{C}^n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_15352 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Every nonflat conformal minimal surface is homotopic to a proper one Vrhovnik, Tjasa Differential Geometry Complex Variables Given an open Riemann surface $M$, we prove that every nonflat conformal minimal immersion $M\to\mathbb{R}^n$ ($n\geq 3$) is homotopic through nonflat conformal minimal immersions $M\to\mathbb{R}^n$ to a proper one. If $n\geq 5$, it may be chosen in addition injective, hence a proper conformal minimal embedding. Prescribing its flux, as a consequence, every nonflat conformal minimal immersion $M\to\mathbb{R}^n$ is homotopic to the real part of a proper holomorphic null embedding $M\to\mathbb{C}^n$. We also obtain a result for a more general family of holomorphic immersions from an open Riemann surface into $\mathbb{C}^n$ directed by Oka cones in $\mathbb{C}^n$. |
| title | Every nonflat conformal minimal surface is homotopic to a proper one |
| topic | Differential Geometry Complex Variables |
| url | https://arxiv.org/abs/2505.15352 |