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Deformation and singularities of maximal surfaces with planar curvature lines. (English) Zbl 1396.53081
Summary: Minimal surfaces with planar curvature lines in the Euclidean space have been studied since the late nineteenth century. On the other hand, the classification of maximal surfaces with planar curvature lines in the Lorentz-Minkowski space has only recently been given. In this paper, we use an alternative method not only to refine the classification of maximal surfaces with planar curvature lines, but also to show that there exists a deformation consisting exactly of all such surfaces. Furthermore, we investigate the types of singularities that occur on maximal surfaces with planar curvature lines. Finally, by considering the conjugate of maximal surfaces with planar curvature lines, we obtain analogous results for maximal surfaces that are also affine minimal surfaces.

MSC:
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
58K05 Critical points of functions and mappings on manifolds
53B25 Local submanifolds
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