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Maximal surfaces with conelike singularities of finite type. (English) Zbl 1009.53046

If \(\psi:M\to L^3\) describes an oriented spacelike surface in the 3-dimensional Minkowski space \(L^3\), then its unit normal field \(\nu\) takes its values in the hyperbolic plane \(H^2\subset L^3\), which by the “hyperbolic stereographic projection” \(\sigma\) is mapped onto the unit disc \(\Delta\subset \mathbb{C}\). The Gauss map of \(\psi\) is defined by \(g:=\sigma \circ\nu\). If \(\psi\) is maximal (i.e., its mean curvature vanishes), then \(g\) is a holomorphic function on the Riemann surface \(M\); the only complete maximal examples are planes. In 1983/84 O. Kobayashi has introduced the concept of conelike singularities. He modified the notion of completeness and has shown that the only complete maximal example with a conelike singularity and an injective Gauss map \(g\) is the Lorentzian catenoid. In the paper under review this result is improved by replacing the injectivity of \(g\) by some condition on the behaviour of \(g\) “along the conelike singularities”.

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
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