Eventual Monotonicity of the Willmore Energy under Mean Curvature Flow

RESUMO

We prove a late-time monotonicity of the Willmore deficit \(\int_{\Sigma}H^2\,d\mu-16\pi\) 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.