On Minimal Surfaces of Revolutions Immersed in Deformed Hyperbolic Kropina Space

In this paper we consider three dimensional upper half space ${\mathbb{H}}^3$ equipped with various Kropina metrics obtained by deformation of hyperbolic metric of ${\mathbb{H}}^3$ through $1$-forms and obtain a partial differential equation that characterizes minimal surfaces immersed in it. We prove that such minimal surfaces can only be obtained when the hyperbolic metric is deformed along $x^3$ direction. Then we classify such minimal surfaces and show that flag curvature of these surfaces is always non-positive. We also obtain the geodesics of this surface. In particular, it follows that such surfaces neither have forward conjugate points nor they are forward complete.