Singular minimal translation graphs in Euclidean spaces

J. Korean Math. Soc. 2021 Vol. 58, No. 1, 109-122 https://doi.org/10.4134/JKMS.j190851 Published online December 2, 2020 Printed January 1, 2021

Muh{$\dot{\textsc i}$}tt{$\dot{\textsc i}$}n Evren Aydin, Ayla Erdur, Mahmut Erg\"ut Firat University; Tekirdag Namik Kemal University; Tekirdag Namik Kemal University

Abstract : In this paper, we consider the problem of finding the hypersurface $M^{n}$ in the Euclidean $\left( n+1\right)$-space $\mathbb{R}^{n+1}$ that satisfies an equation of mean curvature type, called singular minimal hypersurface equation. Such an equation physically characterizes the surfaces in the upper halfspace $\mathbb{R}_{+}^{3}\left( \mathbf{u} \right) $ with lowest gravity center, for a fixed unit vector $\mathbf{u}\in \mathbb{R}^{3}$. We first state that a singular minimal cylinder $M^{n}$ in $\mathbb{R}^{n+1}$ is either a hyperplane or a $\alpha $-catenary cylinder. It is also shown that this result remains true when $M^{n}$ is a translation hypersurface and $\mathbf{u}$ is a horizantal vector. As a further application, we prove that a singular minimal translation graph in $\mathbb{R }^{3}$ of the form $z=f(x)+g(y+cx),$ $c\in \mathbb{R-\{}0\},$ with respect to a certain horizantal vector $\mathbf{u}$ is either a plane or a $\alpha $- catenary cylinder.