Quantitative Stability of the Clifford Torus as a Willmore Minimizer

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 $$\mathcal{W}(V)\le 2{\pi }^2+{\delta }^2,\delta <{\delta }_0\ll 1.$$ We show that the support $\mathrm{\Sigma }=\mathrm{spt}V$ 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 $\mathrm{\Sigma }$ 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 $C\delta$.