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On surfaces of finite total curvature. (English) Zbl 0853.53003

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.
Reviewer: O.Röschel (Graz)

MSC:

53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
53A30 Conformal differential geometry (MSC2010)
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