A genus-zero surface with bounded curvature enclosing less volume than the unit sphere
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866917163967709184 |
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| author | Bolan, Matthew |
| author_facet | Bolan, Matthew |
| contents | We produce a family of bodies in $\mathbb R^3$ parameterized by $\varepsilon > 0$, each bounded by a smooth topological sphere with principal curvatures in $[-1, 1]$, and having volume arbitrarily close to $ 16 - 4\sqrt 3 + \left(10 \sqrt 3 - 14\right) π- \left(\frac{10}{3} - \sqrt 3\right) π^2 \approx 3.70.$ Thus, in contrast to the two-dimensional case, the unit sphere (which bounds a ball of volume $\frac{4 }{ 3} π\approx 4.19$) does not enclose the minimal volume among all smooth spheres in $\mathbb R^3$ with principal curvatures in $[-1,1]$. This answers a folklore question of Dmitri Burago and Anton Petrunin. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_19659 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A genus-zero surface with bounded curvature enclosing less volume than the unit sphere Bolan, Matthew Differential Geometry We produce a family of bodies in $\mathbb R^3$ parameterized by $\varepsilon > 0$, each bounded by a smooth topological sphere with principal curvatures in $[-1, 1]$, and having volume arbitrarily close to $ 16 - 4\sqrt 3 + \left(10 \sqrt 3 - 14\right) π- \left(\frac{10}{3} - \sqrt 3\right) π^2 \approx 3.70.$ Thus, in contrast to the two-dimensional case, the unit sphere (which bounds a ball of volume $\frac{4 }{ 3} π\approx 4.19$) does not enclose the minimal volume among all smooth spheres in $\mathbb R^3$ with principal curvatures in $[-1,1]$. This answers a folklore question of Dmitri Burago and Anton Petrunin. |
| title | A genus-zero surface with bounded curvature enclosing less volume than the unit sphere |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2512.19659 |