Müller, S.; Šverák, V. On surfaces of finite total curvature. (English) Zbl 0853.53003 J. Differ. Geom. 42, No. 2, 229-258 (1995). 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) Cited in 3 ReviewsCited in 54 Documents 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) Keywords:surfaces; finite total curvature; embeddings; proper immersions; conformal parametrization × Cite Format Result Cite Review PDF Full Text: DOI