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Channel linear Weingarten surfaces in space forms

Udo Hertrich-Jeromin, Mason Pember, Denis Polly

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.

Channel linear Weingarten surfaces in space forms

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.

Paper Structure

This paper contains 28 sections, 18 theorems, 142 equations, 8 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Linear Weingarten surfaces
  3. Lie sphere geometry and its subgeometries
  4. Surfaces
  5. Curvature
  6. Rotational surfaces
  7. Non-isotropic cases
  8. Isotropic cases
  9. Parabolic rotational surfaces in H3
  10. Surfaces of revolution in R3
  11. Channel linear Weingarten surfaces
  12. Constant Gauss curvature
  13. Non-isotropic cases
  14. Positive intrinsic Gauss curvature.
  15. Negative extrinsic Gauss curvature.
  16. ...and 13 more sections

Key Result

Lemma 2.1

Given a point sphere complex $\mathfrak{p}$ and a Legendre immersion $\Lambda$, there is precisely one point sphere congruence $\mathfrak{f}$ enveloped by $\Lambda$.

Figures (8)

  • Figure 1: Visualisation of subgroups of rotation in the hyperbolic plane: The orbits of one point under all three types of subgroup of rotation are shown in the Poincaré disk and the half plane model.
  • Figure 2: Profile curves of CGC surfaces of elliptic rotation in $\mathbb{S}^3$ after stereographic projection: the dashed line represents the axis of rotation. The profile curves are obtained from the solutions in \ref{['tab:SphericalCase']} and the surfaces obtained via rotation are displayed in \ref{['fig:SphereSurfaces']}.
  • Figure 3: Stereographic projections of rotational surfaces of different Gauss curvatures in $\mathbb{S}^3$. The figures are obtained from the parametrisation given in \ref{['thm:Sphere']}, with the functions $r$ and $\psi$ from \ref{['tab:SphericalCase']}; the meridian curves are shown in Figure \ref{['fig:Curves_S3']}.
  • Figure 4: Profile curves of CGC surfaces of elliptic rotation in the half plane model: the dashed line represents the axis of rotation. The profile curves are obtained using the solutions in \ref{['tab:HyperbolicCaseElliptic']}. The elliptic rotational surfaces corresponding to the profile curves in Figures \ref{['fig:Curves_H3_ell_NegCN']}, \ref{['fig:Curves_H3_ell_MixSC']} and \ref{['fig:Curves_H3_ell_PosCN']} are displayed in Figures \ref{['fig:Ell_Neg']}, \ref{['fig:Ell_Mix']} and \ref{['fig:Ell_Pos']}, respectively.
  • Figure 5: The function $P$ in \ref{['eq:Period']} is continuous in $p$ and, for suitably large $n\in \mathbb{N}$, greater than $2\pi$ at $0$ with a zero at $p=\sqrt{\tfrac{1}{1-K}}$ (here $K=-1$). Thus, by the intermediate value theorem, there is a unique solution $p$ of \ref{['eq:Period']}.
  • ...and 3 more figures

Theorems & Definitions (41)

  • Lemma 2.1
  • Definition 2.2
  • Remark 2.3
  • Proposition 2.4
  • Remark 2.5
  • proof
  • Theorem 2.6
  • Remark 2.7
  • Proposition 2.8
  • Remark 2.9
  • ...and 31 more