Singular Minimal Translation Graphs in Euclidean Spaces

TL;DR

This paper studies singular minimal hypersurfaces in Euclidean spaces, characterizing their shapes and proving that certain translation graphs are either planes or specific catenary cylinders, expanding understanding of these geometric structures.

Contribution

It classifies singular minimal hypersurfaces in Euclidean spaces, showing that cylinders and translation graphs of a certain form are either planes or {\

Findings

Singular minimal cylinders are either hyperplanes or {\

Singular minimal translation graphs of a specific form are either planes or {\

Abstract

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