Zero $f$-mean curvature surfaces of revolution in the Lorentzian product ${\mathbb{G}}^2\times {\mathbb{R}}_1$

Abstract

We classify (spacelike or timelike) surfaces of revolution with zero $f$-mean curvature in ${\mathbb{G}}^2\times {\mathbb{R}}_1,$ the Lorentz-Minkowski 3-space ${\mathbb{R}}_1^3$ 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 ${\mathbb{G}}^2\times {\mathbb{R}}_1.$