Three Bernstein type theorems for hypersurfaces with zero Gaussian curvature
Slawomir Dinew, Mengru Guo, Heming Jiao
TL;DR
The paper addresses rigidity for entire convex graphs with zero Gaussian curvature under the degenerate Monge-Ampère equation $\det D^2u=0$ in both Euclidean and Minkowski spaces. It proves Bernstein-type results: if second-order curvature quantities decay at infinity, namely $\lim_{|x|\to\infty}\Delta u=0$ in the Euclidean setting or $\lim_{|x|\to\infty}\tilde{H}^M_u=0$ with no timelike points in Minkowski space, then the graphic hypersurface must be a hyperplane. A counterexample, $u(x)=a\sqrt{x_1^2+c}$ with $0<a<1$, shows that zero Gaussian curvature alone does not enforce affinity. The work situates these results within the broader rigidity theory for Monge-Ampère and Hessian equations, extending classical Bernstein-type conclusions to the degenerate case through convexity arguments and affinity-direction analysis.
Abstract
In this paper, we prove Bernstein type theorems for entire convex graphical hypersurfaces with zero Gaussian curvature in both Euclidean and Minkowski context. A supplementary example illustrates that zero Gaussian convex spacelike hypersurfaces are not necessary hyperplanes without additional conditions. We show that a zero Gaussian curvature convex hypersurface must be a hyperplane if the mean curvature goes to zero at infinity. In the Minkowski context, we prove similar results for hypersurface without timelike points.