# Zero $f$-mean curvature surfaces of revolution in the Lorentzian product $\Bbb G^2\times\Bbb R_1$

## TL;DR

This paper classifies surfaces of revolution with zero $f$-mean curvature in Lorentzian product space, identifying specific types of $f$-maximal and $f$-minimal surfaces, including new examples of $f$-Catenoids.

## Contribution

It provides a complete classification of $f$-mean curvature zero surfaces of revolution in Lorentzian product space, introducing new $f$-Catenoid examples.

## Key findings

- $f$-maximal surfaces are planes or $f$-Catenoids.
- $f$-minimal timelike surfaces are planes, cylinders, or $f$-Catenoids.
- New explicit examples of $f$-Catenoids in Lorentzian space.

## Abstract

We classify (spacelike or timelike) surfaces of revolution with zero $f$-mean curvature in $\Bbb G^2\times\Bbb R_1,$ the Lorentz-Minkowski 3-space $\Bbb R^3_1$ endowed with the Gaussian-Euclidean density $e^{-f(x,y,z)}=\frac 1{2\pi}e^{-\frac{x^2+y^2}2}.$ It is proved that an $f$-maximal surface of revolution is either a horizontal plane or a spacelike $f$-Catenoid. For the timelike case, a timelike $f$-minimal surface is either a vertical plane containing $z$-axis, the cylinder $x^2+y^2=1,$ or a timelike $f$-Catenoid. Spacelike and timelike $f$-Catenoids are new examples of $f$-minimal surfaces in $\Bbb G^2\times \Bbb R_1.$