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 a rescaled branched conformal immersed sequence(c.f. Theorem 1.5). By virtue of this local convergence theorem, we study the blowup behavior of a sequence of branched conformal immersions of closed Riemannian surface in ${\mathbb{R}}^n$ with uniformly bounded areas and Willmore energies. Furthermore, we prove that the integral identity of Gauss curvature is true.