Families of proper minimal surfaces
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866910236562948096 |
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| author | Alarcon, Antonio Forstneric, Franc |
| author_facet | Alarcon, Antonio Forstneric, Franc |
| contents | Assume that $X$ is a connected, open, oriented smooth surface, $B$ is a compact Euclidean neighbourhood retract, and $\mathscr{J}=\{J_b\}_{b\in B}$ is a continuous family of complex structures on $X$ of local Hölder class $\mathscr{C}^α$ for some $0<α<1$. We construct a continuous family of $J_b$-conformal minimal immersions $u_b:X\to \mathbb{R}^3$, $b\in B$, properly projecting to $\mathbb{R}^2$ and having an arbitrary given family of flux homomorphisms ${\rm Flux}_{u_b}:H_1(X,\mathbb{Z})\to\mathbb{R}^3$. In particular, there are continuous families of proper $J_b$-holomorphic null immersions $X\to \mathbb{C}^3$ and of proper $J_b$-holomorphic immersions $X\to\mathbb{C}^2$, $b\in B$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_19883 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Families of proper minimal surfaces Alarcon, Antonio Forstneric, Franc Differential Geometry Complex Variables Assume that $X$ is a connected, open, oriented smooth surface, $B$ is a compact Euclidean neighbourhood retract, and $\mathscr{J}=\{J_b\}_{b\in B}$ is a continuous family of complex structures on $X$ of local Hölder class $\mathscr{C}^α$ for some $0<α<1$. We construct a continuous family of $J_b$-conformal minimal immersions $u_b:X\to \mathbb{R}^3$, $b\in B$, properly projecting to $\mathbb{R}^2$ and having an arbitrary given family of flux homomorphisms ${\rm Flux}_{u_b}:H_1(X,\mathbb{Z})\to\mathbb{R}^3$. In particular, there are continuous families of proper $J_b$-holomorphic null immersions $X\to \mathbb{C}^3$ and of proper $J_b$-holomorphic immersions $X\to\mathbb{C}^2$, $b\in B$. |
| title | Families of proper minimal surfaces |
| topic | Differential Geometry Complex Variables |
| url | https://arxiv.org/abs/2605.19883 |