A survey on Bernstein-type theorems for entire graphical surfaces
arXiv:2403.14199
Abstract
We survey Bernstein-type theorems for graphical surfaces in the Euclidean space and the Lorentz-Minkowski space. More specifically, we explain several proofs of the Bernstein theorem for minimal graphs in the Euclidean 3-space. Furthermore, we show the Heinz-type mean curvature estimates for graphs in the Euclidean 3-space and space-like graphs in the Lorentz-Minkowski 3-space. As an application of these estimates, we give Bernstein-type theorems for constant mean curvature graphs in the Euclidean 3-space and constant mean curvature space-like graphs in the Lorentz-Minkowski 3-space, respectively. We also study Bernstein-type results for minimal graphs in the Euclidean 4-space and the Calabi-Bernstein theorem in the Lorentz-Minkowski 3-space.
13 pages, No figure, correcting some typos and updating references, to appear in the ASPM volume of the 13th MSJ-SI proceedings "Differential Geometry and Integrable Systems"