In this presentation, we talk about the stability of the Willmore functional. For an integral 2-varifold $V=\underline{v}(\Sigma,\theta)$ in $R^n$ with square integrable generalized mean curvature and finite mass. If its Willmore energy is smaller than $4\pi(1+\delta^2)$ and the mass is normalized to be $4\pi$, we show that $\Sigma$ is $W^{2,2}$ and bi-Lipschitz close to the round sphere in a quantitative way when $\delta<\delta_0\ll1$. For $n=3$, we show the sharp constant is $\delta_0^2=2\pi$. This is a joint work with Dr. Yuchen Bi at Peking University.
