Bernstein-Type Theorem for Zero Mean Curvature Hypersurfaces Without Time-like Points in Lorentz–Minkowski Space

Abstract

Calabi and Cheng-Yau’s Bernstein-type theorem asserts that an entire zero mean curvature graph in Lorentz–Minkowski $$(n+1)$$ -space $${{\varvec{R}}_{1}^{n+1}}$$ 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 $${\varvec{R}}^{n+1}_1$$ 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 $$n=2$$ .