# Helicoids and Catenoids in $M\times\mathbb{R}$

## TL;DR

This paper explores the construction and properties of minimal hypersurfaces called vertical helicoids and catenoids in product manifolds, extending classical Euclidean examples to more general Riemannian manifolds.

## Contribution

It introduces the concepts of vertical helicoids and catenoids in arbitrary Riemannian manifolds and characterizes their existence and uniqueness properties, extending classical minimal surface theory.

## Key findings

- Vertical helicoids have similar uniqueness properties as in Euclidean space.
- Vertical catenoids exist iff the manifold admits isoparametric hypersurfaces.
- Complete classification of constant angle hypersurfaces 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 vertical helicoids in the case where $M$ is one of the simply connected space forms. Vertical helicoids which are entire graphs of functions on ${\rm Nil}_3$ and ${\rm Sol}_3$ are also presented. 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, they 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.