Discrete improper affine spheres.
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)].
scientific article; zbMATH DE number 1886818