Non-extendability of complex structures
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866908754771968000 |
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| author | Tang, Zizhou Yan, Wenjiao |
| author_facet | Tang, Zizhou Yan, Wenjiao |
| contents | There exists a complex structure $J$ on a connected open subset $S^3_δ\times S^3$ of $S^6$. The present paper proves that: (1) $J$ can be extended to a global almost complex structure $\widetilde{J}$ on $S^6$; (2) any extension to $S^6$ is necessarily non-integrable. Therefore, it is impossible to deform $\widetilde{J}$ to an integrable almost complex structure on $S^6$ while fixing it on $S^3_δ\times S^3$. This phenomenon indicates that the deformation strategy suggested by S.-T. Yau in his Problem 52 cannot be realized in this sense. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_05568 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Non-extendability of complex structures Tang, Zizhou Yan, Wenjiao Complex Variables Algebraic Topology Differential Geometry 32G05, 32Q60, 53C15 There exists a complex structure $J$ on a connected open subset $S^3_δ\times S^3$ of $S^6$. The present paper proves that: (1) $J$ can be extended to a global almost complex structure $\widetilde{J}$ on $S^6$; (2) any extension to $S^6$ is necessarily non-integrable. Therefore, it is impossible to deform $\widetilde{J}$ to an integrable almost complex structure on $S^6$ while fixing it on $S^3_δ\times S^3$. This phenomenon indicates that the deformation strategy suggested by S.-T. Yau in his Problem 52 cannot be realized in this sense. |
| title | Non-extendability of complex structures |
| topic | Complex Variables Algebraic Topology Differential Geometry 32G05, 32Q60, 53C15 |
| url | https://arxiv.org/abs/2601.05568 |