Holomorphicity of parabolic stable minimal surfaces of high codimension
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866915516395814912 |
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| author | Cheng, Da Rong Karigiannis, Spiro Madnick, Jesse |
| author_facet | Cheng, Da Rong Karigiannis, Spiro Madnick, Jesse |
| contents | A classical theorem of Micallef says that if $F \colon (Σ, g) \to \mathbb{R}^4$ is a stable minimal immersion of an oriented $2$-dimensional complete Riemannian manifold (that is parabolic) into $\mathbb{R}^4$, it is necessarily holomorphic with respect to some parallel orthogonal complex structure on $\mathbb{R}^4$. We generalize this theorem by replacing $\mathbb{R}^4$ with $\mathbb{R}^{2 + 2k}$ for any codimension $2k$, under the additional hypothesis that the normal bundle $N Σ$ is equipped with a complex structure that is compatible with the induced metric and parallel with respect to the induced connection. This is a necessary assumption for such a theorem to hold, and it is automatically satisfied in the classical case $k=1$. We also briefly discuss possible further generalizations of such a result to other calibrations and to Smith maps. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_22155 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Holomorphicity of parabolic stable minimal surfaces of high codimension Cheng, Da Rong Karigiannis, Spiro Madnick, Jesse Differential Geometry 53C15, 53C38, 53C42 A classical theorem of Micallef says that if $F \colon (Σ, g) \to \mathbb{R}^4$ is a stable minimal immersion of an oriented $2$-dimensional complete Riemannian manifold (that is parabolic) into $\mathbb{R}^4$, it is necessarily holomorphic with respect to some parallel orthogonal complex structure on $\mathbb{R}^4$. We generalize this theorem by replacing $\mathbb{R}^4$ with $\mathbb{R}^{2 + 2k}$ for any codimension $2k$, under the additional hypothesis that the normal bundle $N Σ$ is equipped with a complex structure that is compatible with the induced metric and parallel with respect to the induced connection. This is a necessary assumption for such a theorem to hold, and it is automatically satisfied in the classical case $k=1$. We also briefly discuss possible further generalizations of such a result to other calibrations and to Smith maps. |
| title | Holomorphicity of parabolic stable minimal surfaces of high codimension |
| topic | Differential Geometry 53C15, 53C38, 53C42 |
| url | https://arxiv.org/abs/2509.22155 |