Eventual Monotonicity of the Willmore Energy under Mean Curvature Flow

We prove a late-time monotonicity of the Willmore deficit ${\int }_{\mathrm{\Sigma }}{H}^{2}d\mu -16\pi$ along the mean curvature flow of smooth closed embedded strictly convex surfaces in ${\mathbb{R}}^3$,which refining the exponential convergence given by Huisken. As a consequence, the Hawking mass is eventually non–decreasing along mean curvature flow. We also discuss totally umbilical solutions in three-dimensional space forms, where the monotonicity directions of the Willmore energy and the Hawking mass depend on the sign of the ambient sectional curvature.