Linear Quantitative Rigidity for Almost-CMC Surfaces

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.