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In this paper, we first extend the classical Hélein’s convergence theorem to a sequence of rescaled branched conformal immersions. By virtue of this local convergence theorem, we study the blow-up behavior of a sequence of branched conformal immersions of a closed Riemann surface in ℝn with uniformly bounded areas and Willmore energies. Furthermore, we prove that the integral identity of Gauss curvature is true.

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