Helicoids and catenoids in $M\times \mathbb {R}$

Abstract

Given an arbitrary $C^\infty$ Riemannian manifold $M^n$, we consider the problem of introducing and constructing minimal hypersurfaces in $M\times \mathbb {R}$ which have the same fundamental properties of the standard helicoids and catenoids of Euclidean space $\mathbb {R}^3=\mathbb {R} ^2\times \mathbb {R}$. Such hypersurfaces are defined by imposing conditions on their height functions and horizontal sections and then called vertical helicoids and vertical catenoids. We establish that vertical helicoids in $M\times \mathbb {R}$ have the same fundamental uniqueness properties of the helicoids in $\mathbb {R}^3.$ We provide several examples of properly embedded vertical helicoids in the case where M is one of the simply connected space forms. Vertical helicoids which are entire graphs of functions on $\mathrm{Nil}_3$ and $\mathrm{Sol}_3$ are also presented. We show that vertical helicoids of $M\times \mathbb {R}$ whose horizontal sections are totally geodesic in M are locally given by a “twisting” of a fixed totally geodesic hypersurface of M. We give a local characterization of hypersurfaces of $M\times \mathbb {R}$ which have the gradient of their height functions as a principal direction. As a consequence, we prove that vertical catenoids exist in $M\times \mathbb {R}$ if and only if M admits families of isoparametric hypersurfaces. If so, properly embedded vertical catenoids can be constructed through the solutions of a certain first-order linear differential equation. Finally, we give a complete classification of the hypersurfaces of $M\times \mathbb {R}$ whose angle function is constant.