Linear Quantitative Rigidity for Almost--CMC Surfaces

We prove a quantitative rigidity result for almost constant mean curvature spheres in 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 ∞ -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.