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

TL;DR

This paper extends a key convergence theorem for branched conformal immersions, analyzes their blowup behavior under bounded geometric energies, and confirms the Gauss curvature integral identity.

Contribution

It generalizes Hélain's theorem to rescaled sequences and studies the blowup behavior of branched conformal immersions with bounded energies.

Findings

Extended Hélain's theorem to rescaled sequences

Analyzed blowup behavior of conformal immersions

Confirmed Gauss curvature integral identity

Abstract

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.