Linear Quantitative Rigidity for Almost-CMC Surfaces
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866908765123510272 |
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| author | Bi, Yuchen Zhou, Jie |
| author_facet | Bi, Yuchen Zhou, Jie |
| contents | We prove a quantitative rigidity result for almost constant mean curvature spheres in $\mathbb{R}^3$. Under a sub--two--sphere Willmore bound and a small $L^2$--CMC defect, we show that an almost--CMC surface is close to the round sphere, with linear control of the $W^{2,2}$--distance of the parametrization and the $L^\infty$--norm of the conformal factor. An analogous statement holds under an a priori area bound below that of two spheres.The proof relies on a linearized analysis around the sphere. A previously established qualitative rigidity result provides the initial closeness required to enter the perturbative regime. The estimate further extends to integral $2$--varifolds of unit density using known regularity and density results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_09457 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Linear Quantitative Rigidity for Almost-CMC Surfaces Bi, Yuchen Zhou, Jie Differential Geometry We prove a quantitative rigidity result for almost constant mean curvature spheres in $\mathbb{R}^3$. Under a sub--two--sphere Willmore bound and a small $L^2$--CMC defect, we show that an almost--CMC surface is close to the round sphere, with linear control of the $W^{2,2}$--distance of the parametrization and the $L^\infty$--norm of the conformal factor. An analogous statement holds under an a priori area bound below that of two spheres.The proof relies on a linearized analysis around the sphere. A previously established qualitative rigidity result provides the initial closeness required to enter the perturbative regime. The estimate further extends to integral $2$--varifolds of unit density using known regularity and density results. |
| title | Linear Quantitative Rigidity for Almost-CMC Surfaces |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2601.09457 |