Singular Minimal Translation Graphs in Euclidean Spaces

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 α-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 α-catenary cylinder.