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 language | English |
|---|---|
| Pages (from-to) | 969-1009 |
| Number of pages | 41 |
| Journal | Beitrage zur Algebra und Geometrie |
| Volume | 64 |
| Issue number | 4 |
| Early online date | 23 Jan 2023 |
| DOIs | |
| Publication status | Published - 31 Dec 2023 |
Funding
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. 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.
| Funders | Funder number |
|---|---|
| Feray Bayar | |
| Istituto Nazionale di Alta Matematica "Francesco Severi" | |
| Politecnico di Torino | |
| Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni | |
| Japan Society for the Promotion of Science | P28427-N35, I3809-N32 |
| Austrian Science Fund | |
| Ministero dell’Istruzione, dell’Università e della Ricerca | E11G18000350001 |
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