Generalized isothermic lattices

Abstract

We study multidimensional quadrilateral lattices satisfying simultaneously two integrable constraints: a quadratic constraint and the projective Moutard constraint. When the lattice is two dimensional and the quadric under consideration is the Moebius sphere one obtains, after the stereographic projection, the discrete isothermic surfaces defined by Bobenko and Pinkall by an algebraic constraint imposed on the (complex) cross-ratio of the circular lattice. We derive the analogous condition for our generalized isthermic lattices using Steiner's projective structure of conics and we present basic geometric constructions which encode integrability of the lattice. In particular, we introduce the Darboux transformation of the generalized isothermic lattice and we derive the corresponding Bianchi permutability principle. Finally, we study two dimensional generalized isothermic lattices, in particular geometry of their initial boundary value problem.

19 pages, 11 figures; v2. some typos corrected; v3. new references added, higlighted similarities and differences with recent papers on the subject

References in corpus (1)

• The B-quadrilateral lattice, its transformations and the algebro-geometric construction

Cited by in corpus (5)

• Discrete linear Weingarten surfaces
• Non-commutative lattice modified Gel'fand-Dikii systems
• Integrable lattices and their sublattices II. From the B-quadrilateral lattice to the self-adjoint schemes on the triangular and the honeycomb lattices
• Quadrangular sets in projective line and in Moebius space, and geometric interpretation of the non-commutative discrete Schwarzian Kadomtsev-Petviashvili equation
• Covariant description of isothermic surfaces