Weierstrass representation for surfaces in the three-dimensional Heisenberg group

2009 ◽  
Vol 31 (1) ◽  
pp. 119-132
Author(s):  
Qun Chen ◽  
Hongbing Qiu
Keyword(s):  
Heisenberg Group ◽  
Three Dimensional ◽  
Weierstrass Representation

scholarly journals Einstein–Weyl geometry, dispersionless Hirota equation and Veronese webs

2014 ◽  
Vol 157 (1) ◽  
pp. 139-150 ◽  
Author(s):  
MACIEJ DUNAJSKI ◽  
WOJCIECH KRYŃSKI
Keyword(s):  
Heisenberg Group ◽  
Twistor Space ◽  
Three Dimensional ◽  
Poisson Structures ◽  
Weyl Geometry ◽  
Hirota Equation ◽  
Five Dimensions ◽  
Dispersionless Hirota Equation

AbstractWe exploit the correspondence between the three–dimensional Lorentzian Einstein–Weyl geometries of the hyper–CR type and the Veronese webs to show that the former structures are locally given in terms of solutions to the dispersionless Hirota equation. We also demonstrate how to construct hyper–CR Einstein–Weyl structures by Kodaira deformations of the flat twistor space$T\mathbb{CP}^1$, and how to recover the pencil of Poisson structures in five dimensions illustrating the method by an example of the Veronese web on the Heisenberg group.

scholarly journals Ruled minimal surfaces in the three-dimensional Heisenberg group

2013 ◽  
Vol 261 (2) ◽  
pp. 477-496 ◽  
Author(s):  
Heayong Shin ◽  
Young Wook Kim ◽  
Sung-Eun Koh ◽  
Hyung Yong Lee ◽  
Seong-Deog Yang
Keyword(s):  
Heisenberg Group ◽  
Minimal Surfaces ◽  
Three Dimensional

Surfaces in the three-dimensional Heisenberg group on which the Gauss map has bounded Jacobian

2011 ◽  
Vol 89 (5-6) ◽  
pp. 746-748 ◽  
Author(s):  
A. A. Borisenko ◽  
E. V. Petrov
Keyword(s):  
Heisenberg Group ◽  
Three Dimensional ◽  
Gauss Map

scholarly journals A loop group method for minimal surfaces in the three-dimensional Heisenberg group

2016 ◽  
Vol 20 (3) ◽  
pp. 409-448
Author(s):  
Josef F. Dorfmeister ◽  
Jun-Ichi Inoguchi ◽  
Shimpei Kobayashi
Keyword(s):  
Heisenberg Group ◽  
Minimal Surfaces ◽  
Three Dimensional ◽  
Loop Group ◽  
Group Method

scholarly journals TSALLIS ENTROPY COMPOSITION AND THE HEISENBERG GROUP

2013 ◽  
Vol 10 (07) ◽  
pp. 1350032 ◽  
Author(s):  
NIKOS KALOGEROPOULOS
Keyword(s):  
Phase Space ◽  
Euclidean Space ◽  
Heisenberg Group ◽  
Polynomial Growth ◽  
Three Dimensional ◽  
Tsallis Entropy ◽  
Geometric Framework ◽  
Fractal Properties ◽  
Composition Property ◽  
Riemannian Spaces

We present an embedding of the Tsallis entropy into the three-dimensional Heisenberg group, in order to understand the meaning of generalized independence as encoded in the Tsallis entropy composition property. We infer that the Tsallis entropy composition induces fractal properties on the underlying Euclidean space. Using a theorem of Milnor/Wolf/Tits/Gromov, we justify why the underlying configuration/phase space of systems described by the Tsallis entropy has polynomial growth for both discrete and Riemannian cases. We provide a geometric framework that elucidates Abe's formula for the Tsallis entropy, in terms the Pansu derivative of a map between sub-Riemannian spaces.

scholarly journals The Chabauty space of closed subgroups of the three-dimensional Heisenberg group

2009 ◽  
Vol 240 (1) ◽  
pp. 1-48 ◽  
Author(s):  
Martin R. Bridson ◽  
Pierre de la Harpe ◽  
Victor Kleptsyn
Keyword(s):  
Heisenberg Group ◽  
Three Dimensional

scholarly journals Quotients of degenerate Sklyanin algebras

2017 ◽  
Vol 16 (05) ◽  
pp. 1750085 ◽  
Author(s):  
Kevin De Laet
Keyword(s):  
Heisenberg Group ◽  
Hilbert Series ◽  
Three Dimensional ◽  
Central Element ◽  
Sklyanin Algebras ◽  
Sklyanin Algebra

In this paper, it is shown how the Heisenberg group of order 27 can be used to construct quotients of degenerate Sklyanin algebras. These quotients have properties similar to the classical Sklyanin case in the sense that they have the same Hilbert series, the same character series and a central element of degree 3. Regarding the central element of a three-dimensional Sklyanin algebra, a better way to view this using Heisenberg-invariants is shown.

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