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.