A dual simple proof of the classical Bernstein and Calabi–Bernstein theorems

Abstract

In this note, we present a short and simple proof of both the Bernstein theorem and the Calabi–Bernstein theorem, which allows us to visualize both the common features and the remarkable differences between the two results. The proofs are based on the construction of conformal metrics that ensure the parabolicity of the surfaces. Consequently, they do not rely on complex analysis.