Isotopies of complete minimal surfaces of finite total curvature
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| author | Alarcon, Antonio Forstneric, Franc Larusson, Finnur |
| author_facet | Alarcon, Antonio Forstneric, Franc Larusson, Finnur |
| contents | Let $M$ be a Riemann surface biholomorphic to an affine algebraic curve. We show that the inclusion of the space $\Re \mathrm{NC}_*(M,\mathbb{C}^n)$ of real parts of nonflat proper algebraic null immersions $M\to\mathbb{C}^n$, $n\ge 3$, into the space $\mathrm{CMI}_*(M,\mathbb{R}^n)$ of complete nonflat conformal minimal immersions $M\to\mathbb{R}^n$ of finite total curvature is a weak homotopy equivalence. We also show that the $(1,0)$-differential $\partial$, mapping $\mathrm{CMI}_*(M,\mathbb{R}^n)$ or $\Re \mathrm{NC}_*(M,\mathbb{C}^n)$ to the space $\mathscr{A}^1(M,\mathbf{A})$ of algebraic $1$-forms on $M$ with values in the punctured null quadric $\mathbf{A} \subset \mathbb{C}^n\setminus\{0\}$, is a weak homotopy equivalence. Analogous results are obtained for proper algebraic immersions $M\to\mathbb{C}^n$, $n\ge 2$, directed by a flexible or algebraically elliptic punctured cone in $\mathbb{C}^n\setminus\{0\}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_04767 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Isotopies of complete minimal surfaces of finite total curvature Alarcon, Antonio Forstneric, Franc Larusson, Finnur Differential Geometry Complex Variables Primary 53A10. Secondary 30F99, 32E30, 32H02, 32Q56, 53C42 Let $M$ be a Riemann surface biholomorphic to an affine algebraic curve. We show that the inclusion of the space $\Re \mathrm{NC}_*(M,\mathbb{C}^n)$ of real parts of nonflat proper algebraic null immersions $M\to\mathbb{C}^n$, $n\ge 3$, into the space $\mathrm{CMI}_*(M,\mathbb{R}^n)$ of complete nonflat conformal minimal immersions $M\to\mathbb{R}^n$ of finite total curvature is a weak homotopy equivalence. We also show that the $(1,0)$-differential $\partial$, mapping $\mathrm{CMI}_*(M,\mathbb{R}^n)$ or $\Re \mathrm{NC}_*(M,\mathbb{C}^n)$ to the space $\mathscr{A}^1(M,\mathbf{A})$ of algebraic $1$-forms on $M$ with values in the punctured null quadric $\mathbf{A} \subset \mathbb{C}^n\setminus\{0\}$, is a weak homotopy equivalence. Analogous results are obtained for proper algebraic immersions $M\to\mathbb{C}^n$, $n\ge 2$, directed by a flexible or algebraically elliptic punctured cone in $\mathbb{C}^n\setminus\{0\}$. |
| title | Isotopies of complete minimal surfaces of finite total curvature |
| topic | Differential Geometry Complex Variables Primary 53A10. Secondary 30F99, 32E30, 32H02, 32Q56, 53C42 |
| url | https://arxiv.org/abs/2406.04767 |