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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2507.06245 |
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| _version_ | 1866915754195025920 |
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| author | Qiu, Hongda |
| author_facet | Qiu, Hongda |
| contents | We prove that if a topological sphere smoothly embedded into $\mathbb{R}^3$ with normal curvatures absolutely bounded by $1$ is contained in an open ball of radius $2$, then the region it bounds must contain a unit ball. This result suggests a potential direction for a problem formulated by D.Burago and A.Petrunin asking whether a topological sphere smoothly embedded in $\mathbb{R}^3$ with normal curvatures absolutely bounded by $1$ encloses a volume of at least $\frac{4}{3}π$. The appendix presents an example illustrating an alternative aspect for this problem. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_06245 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the curvature bounded sphere problem in $\mathbb{R}^3$ Qiu, Hongda Differential Geometry We prove that if a topological sphere smoothly embedded into $\mathbb{R}^3$ with normal curvatures absolutely bounded by $1$ is contained in an open ball of radius $2$, then the region it bounds must contain a unit ball. This result suggests a potential direction for a problem formulated by D.Burago and A.Petrunin asking whether a topological sphere smoothly embedded in $\mathbb{R}^3$ with normal curvatures absolutely bounded by $1$ encloses a volume of at least $\frac{4}{3}π$. The appendix presents an example illustrating an alternative aspect for this problem. |
| title | On the curvature bounded sphere problem in $\mathbb{R}^3$ |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2507.06245 |