Minimal surfaces in three-dimensional Matsumoto space

TL;DR

This paper studies minimal surfaces in three-dimensional Matsumoto space, deriving PDEs for such surfaces and proving that the only solutions are planes, both for graph and translation surfaces.

Contribution

It derives PDE characterizations of minimal surfaces in Matsumoto space and proves the uniqueness of planes as minimal surfaces in this setting.

Findings

Planes are the only minimal surfaces that are graphs in Matsumoto space.

Planes are also the only minimal translation surfaces in this space.

The paper provides PDEs characterizing these minimal surfaces.

Abstract

In this paper we consider the Matsumoto metric $F=\frac{\alpha^2}{\alpha-\beta}$, on the three dimensional real vector space and obtain the partial differential equations that characterize the minimal surfaces which are graphs of smooth functions and then we prove that plane is the only such surface. We also obtain the partial differential equation that characterizes the minimal translation surfaces and show that again plane is the only such surface.