Minimal translation hypersurfaces in hyperbolic space

Abstract

In the upper half-space model of the hyperbolic space ${\mathbb{H}}^{n+1}$, a translation hypersurface of type I is a graph $x_{n+1}=f_1(x_1)+\cdots +f_n(x_n)$ and a translation hypersurface of type II is a graph $x_n=f_1(x_1)+\cdots +f_{n-1}(x_{n-1})+g(x_{n+1})$. We classify the minimal translation hypersurfaces of both types for every $n\ge 2$. We obtain these results as particular cases of a classification of the singular minimal translation hypersurfaces of ${\mathbb{R}}^{n+1}$ for an arbitrary exponent $\alpha \ne 0$, with respect to a direction orthogonal (type I) or parallel (type II) to the hyperplane over which the hypersurface is a graph. The minimal translation hypersurfaces of type I are exactly the parabolic cylinders over a one-parameter family of arches, which contradicts a nonexistence theorem in the literature. The minimal translation hypersurfaces of type II are the parabolic cylinders, the vertical totally geodesic hyperplanes and, when $n\ge 3$, the vertical cylinders over the Scherk surface.