Rigidity of Translation Surfaces in the Three-Dimensional Sphere $\mathbb{S}^3$

TL;DR

This paper investigates the rigidity properties of translation surfaces in the 3-sphere, describing their geometry through generating curves and establishing a correspondence with similar surfaces in Euclidean space.

Contribution

It introduces an associated frame for curves in $ ext{S}^3$, characterizes the geometry of translation surfaces, and establishes a local isometry between surfaces in $ ext{S}^3$ and $ ext{R}^3$.

Findings

Rigidity results for minimal and constant mean curvature surfaces in $ ext{S}^3$.

A new associated frame for curves in $ ext{S}^3$.

A local isometry between translation surfaces in $ ext{S}^3$ and $ ext{R}^3$.

Abstract

A translation surface in the three-dimensional sphere $\mathbb{S}^3$ is a surface generated by the quaternionic product of two curves, called generating curves. In this paper, we present rigidity results for such surfaces. We introduce an associated frame for curves in $\mathbb{S}^3$, and by means of it, we describe the local intrinsic and extrinsic geometry of translation surfaces in $\mathbb{S}^3$. The rigidity results, concerning minimal and constant mean curvature surfaces, are given in terms of the curvature and torsion of the generating curves and their proofs rely on the associated frame of such curves. Finally, we present a correspondence between translation surfaces in $\mathbb{S}^3$ and translation surfaces in $\mathbb{R}^3$. We show that these surfaces are locally isometric, and we present a relation between their mean curvatures.