Quantitative Stability of the Clifford Torus as a Willmore Minimizer
Yuchen Bi, Jie Zhou
TL;DR
The paper proves a sharp quantitative rigidity result forWillmore minimizers in the 3-sphere: any integral $2$-varifold with genus at least one whose Willmore energy is within $δ^2$ of the Clifford torus value $2π^2$ is, after a conformal transformation, $W^{2,2}$-close to a Clifford torus. The authors combine qualitative stability (genus-1 rigidity and convergence to the Clifford torus) with Marques–Neves’ canonical five-parameter family and a minimal-surface stability reduction, then perform a linearized analysis around the Clifford torus to obtain explicit bounds. They show that the conformal factor and the induced metric on the torus vary by at most order $δ$ in $W^{2,2}$ and in the corresponding conformal data, while the mean curvature and area deviate at most linearly in $δ$. This yields a sharp, linear-scale quantitative rigidity result for Willmore minimizers, extending stability theory from genus-zero to genus-one cases on $ olinebreak ext{S}^3$ and highlighting the role of the canonical family in geometric variational analysis.
Abstract
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 $Cδ$.