2019/01/23 by de Lima, Ronaldo F., Roitman, Pedro
#Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.1901.07936
Given an arbitrary C^∞ Riemannian manifold Mn, we consider the problem of introducing and constructing minimal hypersurfaces in M×ℝ which have the same fundamental properties of the standard helicoids and catenoids of Euclidean space ℝ3=ℝ2×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×ℝ have the same fundamental uniqueness properties of the helicoids in ℝ3. 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 Nil3 and \rm Sol3 are also presented. We give a local characterization of hypersurfaces of M×ℝ which have the gradient of their height functions as a principal direction. As a consequence, we prove that vertical catenoids exist in M×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×ℝ whose angle function is constant.