Surfaces in Sol$_3$ space foliated by circles

TL;DR

This paper classifies surfaces in Sol$_3$ space foliated by circles, geodesics, equidistant lines, or horocycles, focusing on their minimality and flatness, and proves the non-existence of zero-curvature surfaces with circle foliations.

Contribution

It provides a complete classification of certain foliated surfaces in Sol$_3$, extending previous work by considering various foliations and curvature conditions.

Findings

No surfaces with zero mean or Gaussian curvature foliated by circles exist in Sol$_3$.

Classified all minimal and flat surfaces foliated by geodesics, equidistant lines, or horocycles.

Extended understanding of surface geometry in Sol$_3$ space.

Abstract

In this paper we study surfaces foliated by a uniparametric family of circles in the homogeneous space Sol$_3$. We prove that there do not exist such surfaces with zero mean curvature or with zero Gaussian curvature. We extend this study considering surfaces foliated by geodesics, equidistant lines or horocycles in totally geodesic planes and we classify all such surfaces under the assumption of minimality or flatness.