A local estimate for maximal surfaces in Lorentzian product spaces

TL;DR

This paper develops a local integral inequality for maximal surfaces in Lorentzian product spaces, providing a new proof of the Calabi-Bernstein theorem for such surfaces.

Contribution

It introduces a local approach and integral inequality for maximal surfaces in Lorentzian products, offering an alternative proof of a key theorem.

Findings

Established a local integral inequality for the second fundamental form

Provided an alternative proof of the Calabi-Bernstein theorem

Enhanced understanding of maximal surfaces in Lorentzian product spaces

Abstract

In this paper we introduce a local approach for the study of maximal surfaces immersed into a Lorentzian product space of the form $M^2\times R_1$, where $M^2$ is a connected Riemannian surface and $M^2\times R_1$ is endowed with the product Lorentzian metric. Specifically, we establish a local integral inequality for the squared norm of the second fundamental form of the surface, which allows us to derive an alternative proof of our Calabi-Bernstein theorem given in \cite{AA}.