An Extension of Calabi's Correspondence between the Solutions of Two Bernstein Problems to More General Elliptic Nonlinear Equations

A new correspondence between the solutions of the minimal surface equation in a certain $3$-dimensional Riemannian warped product and the solutions of the maximal surface equation in a $3$-dimensional standard static space-time is given. This widely extends the classical duality between minimal graphs in $3$-dimensional Euclidean space and maximal graphs in $3$-dimensional Lorentz–Minkowski space-time. We highlight the fact that this correspondence can be restricted to the respective classes of entire solutions. As an application, a Calabi–Bernstein-type result for certain static standard space-times is proved.