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| Main Authors: | , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.13064 |
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| _version_ | 1866909695691718656 |
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| author | Lv, Ying Wang, Peng Xie, Zhenxiao |
| author_facet | Lv, Ying Wang, Peng Xie, Zhenxiao |
| contents | In 1985, Bryant stated that a flat $2$-torus admits a minimal isometric immersion into some round sphere if and only if a certain rationality condition is satisfied. We show that the rationality criterion is no longer a necessary, but a sufficient condition for a flat $n$-torus to admit minimal isometric immersions into spheres. We also derive an upper bound for the algebraic irrationality degree of such immersion. This bound is sharp and equals 4 if $n=3$, and explicit embedded examples are provided respectively for each possible degree. A non-homogeneous example is also presented to show that the minimal isometric immersion of flat $n$-tori is no longer necessarily homogeneous when $n\geq 3$. Moreover, we establish a deformation theorem that every flat n-torus admitting a minimal isometric spherical immersion can be isometrically, minimally and homogeneously immersed into a sphere of dimension at most $n^2+n-1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_13064 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Minimal isometric immersions of flat n-tori into spheres Lv, Ying Wang, Peng Xie, Zhenxiao Differential Geometry 53C42, 53C40, 11H06 In 1985, Bryant stated that a flat $2$-torus admits a minimal isometric immersion into some round sphere if and only if a certain rationality condition is satisfied. We show that the rationality criterion is no longer a necessary, but a sufficient condition for a flat $n$-torus to admit minimal isometric immersions into spheres. We also derive an upper bound for the algebraic irrationality degree of such immersion. This bound is sharp and equals 4 if $n=3$, and explicit embedded examples are provided respectively for each possible degree. A non-homogeneous example is also presented to show that the minimal isometric immersion of flat $n$-tori is no longer necessarily homogeneous when $n\geq 3$. Moreover, we establish a deformation theorem that every flat n-torus admitting a minimal isometric spherical immersion can be isometrically, minimally and homogeneously immersed into a sphere of dimension at most $n^2+n-1$. |
| title | Minimal isometric immersions of flat n-tori into spheres |
| topic | Differential Geometry 53C42, 53C40, 11H06 |
| url | https://arxiv.org/abs/2504.13064 |