Channel linear Weingarten surfaces in space forms

TL;DR

This paper studies a special class of surfaces called channel linear Weingarten surfaces in space forms, showing they are isothermic and often surfaces of revolution, with explicit parametrizations for certain curvature conditions.

Contribution

It provides a unified geometric framework for these surfaces, proving their isothermic property and deriving explicit parametrizations for constant Gauss curvature cases.

Findings

Channel linear Weingarten surfaces are isothermic.

Such surfaces are surfaces of revolution in their ambient space.

Explicit parametrizations are obtained for constant Gauss curvature cases.

Abstract

Channel linear Weingarten surfaces in space forms are investigated in a Lie sphere geometric setting, which allows for a uniform treatment of different ambient geometries. We show that any channel linear Weingarten surface in a space form is isothermic and, in particular, a surface of revolution in its ambient space form. We obtain explicit parametrisations for channel surfaces of constant Gauss curvature in space forms, and thereby for a large class of linear Weingarten surfaces up to parallel transformation.