Bi-Lipschitz rigidity for $L^2$-almost CMC surfaces

For smooth surfaces properly immersed in the unit ball of ${\text{RR}}^n$ with density close to one and small Willmore energy, the optimal a priori estimate(bi-Lipschitz and $W^{2,2}$ parametrization)is provided. We also discuss the quantitative rigidity for $L^2$-almost CMC surfaces.