|
|
Sfb 288 Differential Geometry and Quantum Physics |
Abstract for Sfb Preprint No. 296
Discretization of Surfaces and Integrable Systems A. I. Bobenko and U. Pinkall
In: A.I. Bobenko, R. Seiler (eds.) Discrete Integrable Geometry and Physics, Oxford University Press 1999, pp. 3-58
Discrete analogs of various classes of surfaces and mappings are introduced. These discretizations are characterized by the property that the integrability is preserved, i.e., they are described by discrete integrable systems. As a corollary, rich algebraic structures such as the loop group description, the Baecklund-Darboux transformation, etc. of the corresponding smooth geometries persist in the discrete case. The contribution is a survey of results obtained in this field, and it serves as an introduction to other contributions in that chapter. Combining methods of soliton theory with geometrical intuition, many concrete discrete geometries are described.
Two basic examples are:
discrete K-surfaces (surfaces with constant negative Gaussian curvature),
discrete H-surfaces (surfaces with constant mean curvature),
which are considered in detail.These nets are first derived analytically by discretizing the Lax representation of the corresponding smooth surfaces, preserving the corresponding loop groups. After that, geometrical properties of the nets defined in this way are studied. It is shown that these are natural discrete analogs of geometric properties of the corresponding smooth surfaces.
Other discrete integrable geometries are obtained from generalizations or specializations of these two examples. In all the cases geometrical definitions of the nets are presented. These definitions refer neither to integrable systems nor to the loop group interpretation. For some cases the corresponding Cauchy problems and examples of surfaces are discussed.
Get a gzip-compressed PostScript copy of this preprint
preprint296.ps.gz (224 kB)