Weak convergence of branched conformal immersions with uniformly bounded areas and Willmore energies

In this paper, we firstly extend Theorem 5.1.1 in \cite {Helein} due to H\'elein to the conformal immersions with branched point. By using this local convergence theorem, we study the blowup behavior of a sequence of branched conformal immersions of closed Riemann surface in $\mathbb{R}^{n}$ with uniformly bounded areas and Willmore energies, moreover we prove that the integral of Gauss curvature identity is true.