Discrete improper affine spheres.
type: Item
instance of: scholarly article
title: Discrete improper affine spheres.
zbMATH Open document ID: 1035.53022
DOI: 10.1016/S0393-0440(02)00134-1
author: Nozomu Matsuura
author: Hajime Urakawa
published in: Journal of Geometry and Physics
publication date: 2003-03-23T00:00:00Z
review text: A hypersurface immersion \( f: M \to \mathbb{R}^{n+1} \), \( \dim M = n \), is called an improper affine sphere if it satisfies the equation \( \triangle f = \) const., where \( \triangle \) is the Laplace-Beltrami operator of the Blaschke-Berwald metric \( g \), assuming \( g \) to be regular (definite or indefinite). The paper deals with the dimension \( n = 2 \), emphasizing the indefinite case. Assuming \( g \) in asymptotic parameters \( u,v \), the finding of (non-ruled) improper affine spheres is (save for ODE's) reduced to the Liouville PDE \( w_{uv}+e^{-2w} = 0 \) whose general solution is produced via an integral representation. The main goal is then the discretized version of indefinite improper affine spheres, analogous to the proper case treated by \textit{A. Bobenko} and \textit{W. Schief} [Exp. Math. 8, 261--280 (1999; Zbl 0972.53012)] and [Oxf. Lect. Ser. Math. Appl. 16, 113--138 (1999; Zbl 0942.53013)]. The authors establish the relation between discretized improper affine spheres and \textit{R. Hirota}'s discrete Liouville equation [J. Phys. Soc. Japan 46, 312--319 (1979)]. They also characterize these surfaces by loop groups. In the definite case, the fundamental difference equations and their integrability conditions are given; the Liouville equation then has to be replaced by the PDE \( \triangle w + 4| a| ^2e^{-2w} = 0 \), \( a \) being holomorphic. Additional features might occur if one would utilize the generalized Bäcklund transformations and the integralfree general solutions of these PDEs as introduced by \textit{R. Kirschnick} [Bäcklund transformations and integrability conditions. Dortmund: Univ. Dortmund, Fachbereich Mathematik (1996; Zbl 0999.37051)].
Mathematics Subject Classification ID: 53A15
Mathematics Subject Classification ID: 39A10
Mathematics Subject Classification ID: 35C15
Mathematics Subject Classification ID: 37K10
Mathematics Subject Classification ID: 37K35
zbMATH DE Number: 1886818
zbMATH Keywords: discrete integrable geometry
zbMATH Keywords: affine improper sphere
zbMATH Keywords: Bäcklund transformation
zbMATH Keywords: Liouville equation
zbMATH Keywords: loop group
reviewed by: Rolf Walter
MaRDI profile type: Publication
cites work: Discrete surfaces with constant negative Gaussian curvature and the Hirota equation.
cites work: Affine Spheres: Discretization via Duality Relations
cites work: Q4263406
cites work: Weierstraß-type representation of affine spheres
cites work: Q4846450
cites work: Discretizing constant curvature surfaces via loop group factorizations: The discrete sine- and sinh-Gordon equations
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description: scientific article; zbMATH DE number 1886818