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Bernstein-type theorem for zero mean curvature hypersurfaces without time-like points in Lorentz-Minkowski space. (English) Zbl 1461.53005

Summary: Calabi and Cheng-Yau’s Bernstein-type theorem asserts that an entire zero mean curvature graph in Lorentz-Minkowski \((n+1)\)-space \(\mathbb{R}_1^{n+1}\) which admits only space-like points is a hyperplane. Recently, the third and fourth authors proved a line theorem for hypersurfaces at their degenerate light-like points. Using this, we give an improvement of the Bernstein-type theorem, and we show that an entire zero mean curvature graph in \(\mathbb{R}^{n+1}_1\) consisting only of space-like or light-like points is a hyperplane. This is a generalization of the first, third and fourth authors’ previous result for \(n=2\).

MSC:

53A10 Minimal surfaces in differential geometry, surfaces with prescribed mean curvature
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
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