Singular minimal translation graphs in euclidean spaces

Abstract

In this paper, we consider the problem of finding the hypersurface Mn in the Euclidean (n + 1)-space Rn+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 R3+(u) with lowest gravity center, for a fixed unit vector u?R3 . We first state that a singular minimal cylinder Mn in Rn+1 is either a hyperplane or a ?-catenary cylinder. It is also shown that this result remains true when Mn is a translation hypersurface and u is a horizantal vector. As a further application, we prove that a singular minimal translation graph in R3 of the form z = f(x) + g(y + cx), c ? R ? {0}, with respect to a certain horizantal vector u is either a plane or a ?-catenary cylinder. © 2021 Korean Mathematical Society