Quantitative Stability of the Clifford Torus as a Willmore Minimizer
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909922412724224 |
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| author | Bi, Yuchen Zhou, Jie |
| author_facet | Bi, Yuchen Zhou, Jie |
| contents | For an integral $2$-varifold $V\subset \mathbb{S}^3$ with square-integrable mean curvature, unit density, and support of genus at least $1$, assume that its Willmore energy satisfies \[ \mathcal{W}(V)\le 2π^2+δ^2,\qquad δ<δ_0\ll1. \] We show that the support $Σ=\operatorname{spt}V$ is, after applying a suitable conformal transformation of $\mathbb{S}^3$, quantitatively close to the Clifford torus. More precisely, under an appropriate conformal normalization, the surface $Σ$ admits a $W^{2,2}$ conformal parametrization by the flat torus whose conformal factor and metric coefficients differ from those of the Clifford torus by at most $Cδ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_19681 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantitative Stability of the Clifford Torus as a Willmore Minimizer Bi, Yuchen Zhou, Jie Differential Geometry For an integral $2$-varifold $V\subset \mathbb{S}^3$ with square-integrable mean curvature, unit density, and support of genus at least $1$, assume that its Willmore energy satisfies \[ \mathcal{W}(V)\le 2π^2+δ^2,\qquad δ<δ_0\ll1. \] We show that the support $Σ=\operatorname{spt}V$ is, after applying a suitable conformal transformation of $\mathbb{S}^3$, quantitatively close to the Clifford torus. More precisely, under an appropriate conformal normalization, the surface $Σ$ admits a $W^{2,2}$ conformal parametrization by the flat torus whose conformal factor and metric coefficients differ from those of the Clifford torus by at most $Cδ$. |
| title | Quantitative Stability of the Clifford Torus as a Willmore Minimizer |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2511.19681 |