Channel linear Weingarten surfaces in space forms

Udo Hertrich-Jeromin, Mason Pember, Denis Polly

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish
JournalBeitrage zur Algebra und Geometrie
Early online date23 Jan 2023
DOIs
Publication statusE-pub ahead of print - 23 Jan 2023

Bibliographical note

Funding Information:
The authors would like to thank Feray Bayar, Fran Burstall, Joseph Cho, Shoichi Fujimori, Wayne Rossman and Yuta Ogata for fruitful and helpful discussions. Part of this work was done during a six months stay in Japan, granted to the third author by the FWF/JSPS Joint Project grant I3809-N32 “Geometric shape generation”. Further, this work has been partially supported by the FWF research project P28427-N35 “Non-rigidity and symmetry breaking”. The second author was also supported by GNSAGA of INdAM and the MIUR grant “Dipartimenti di Eccellenza” 2018–2022, CUP: E11G18000350001, DISMA, Politecnico di Torino.

Keywords

  • Channel surface
  • Constant Gauss curvature
  • Isothermic sphere congruence
  • Isothermic surface
  • Jacobi elliptic function
  • Lie sphere geometry
  • Linear Weingarten surface
  • Omega surface

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Geometry and Topology

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