Associated Family and Bäcklund Transformation for Circular K-Nets

Tim Hoffmann, Andrew O'Shea Sageman-Furnas

Description

A $2\times 2$-Lax representation for discrete circular nets of constant negative Gauß curvature is presented here. It is tightly linked to the 4D consistency of the Lax representation of discrete K-nets (in asymptotic line parametrization). The description gives rise to Bäcklund transformations and an associated family. All the members of that family—although no longer circular—can be shown to have constant Gauß curvature as well. Solutions for the Bäcklund transformations of the vacuum (in particular Dini’s surfaces and breather solutions) and their respective associated families are shown as examples.

References

• Tim Hoffmann and Andrew O. Sageman-Furnas.
A $2 \times 2$ Lax Representation, Associated Family, and Bäcklund Transformation for Circular K-Nets.
Discrete & Computational Geometry, 56(2):472–501, September 2016.
arXiv:1510.06654, doi:10.1007/s00454-016-9802-6.

Prof. Dr. Tim Hoffmann   +

Projects: A02
University: TU München, Department of Mathematics, 02.06.021
Address: Boltzmannstr. 3, 85748 Garching, GERMANY
Tel: +49 89 28918384
E-Mail: tim.hoffmann[at]ma.tum.de
Website: https://geo.ma.tum.de/de/personen/tim-hoffmann.html

Dr. Andrew O'Shea Sageman-Furnas   +

Projects: C01
University: TU Berlin, Institut für Mathematik, MA 879
Address: Straße des 17. Juni 136, 10623 Berlin, GERMANY
Tel: +49 30 31429486
E-Mail: aosafu[at]math.tu-berlin.de
Website: https://page.math.tu-berlin.de/~aosafu/