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 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 half- space R-+(3)(u) with lowest gravity center, for a fixed unit vector u is an element of R-3. We first state that a singular minimal cylinder M-n in Rn+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 is 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 is an element of R - {0}, with respect to a certain horizantal vector u is either a plane or a alpha-catenary cylinder.