A connection between flat fronts in hyperbolic space and minimal surfaces in euclidean space

TL;DR

This paper establishes a geometric link between flat fronts in hyperbolic space and minimal surfaces in Euclidean space, extending classical Ribaucour transformation theory to more complex surfaces with umbilic points and topology.

Contribution

It introduces a new construction connecting flat fronts in hyperbolic space with minimal surfaces in Euclidean space, generalizing Ribaucour transformations via complex Riccati equations.

Findings

Provides explicit examples of the construction

Extends Ribaucour transformation theory to non-umbilic and topologically complex surfaces

Reformulates classical theory using complex differential equations

Abstract

A geometric construction is provided that associates to a given flat front in $\mathbb{H}^3$ a pair of minimal surfaces in $\mathbb{R}^3$ which are related by a Ribaucour transformation. This construction is generalized associating to a given frontal in $\mathbb{H}^3$ , a pair of frontals in $\mathbb{R}^3$ that are envelopes of a smooth congruence of spheres. The theory of Ribaucour transformations for minimal surfaces is reformulated in terms of a complex Riccati ordinary differential equation for a holomorphic function. This enables one to simplify and extend the classical theory, that in principle only works for umbilic free and simply connected surfaces, to surfaces with umbilic points and non trivial topology. Explicit examples are included.