On surfaces of finite total curvature
The authors study two-dimensional manifolds \(M\) immersed in \(\mathbb{R}^n\). \(M\) shall be complete, connected, noncompact and oriented. The second fundamental form is denoted by \(A\). Following a result of Huber for the case \(\int_M |A|^2<+ \infty\) there exists a conformal parametrization \(f: S\setminus \{a_1, \dots, a_q \}\to M\to \mathbb{R}^n\) with a compact Riemann surface \(S\). The authors study \(f\) in the neighbourhood of the ends \(a_i\). They show that these immersions are proper and, if \(\int_M |A|^2\leq 4\pi\) or, for \(n=3\), \(\int_M |A|^2< 8\pi\), \(M\) is embedded.
scientific article; zbMATH DE number 894384