Cauchy problems for discrete affine minimal surfaces

Marcos Craizer, Thomas Lewiner, and Ralph Teixeira


M. Craizer and T. Lewinder Department of Mathematics, PUC-Rio, Rua Marquês de São Vicente 255, Gávea, Rio de Janeiro, Brazil,,'

Department of Mathematics, UFF, Niteroi, Brazil

Abstract:In this paper we discuss planar quadrilateral (PQ) nets as discrete models for convex affine surfaces. As a main result, we prove a necessary and sufficient condition for a PQ net to admit a Lelieuvre co-normal vector field. Particular attention is given to the class of surfaces with discrete harmonic co-normals, which we call discrete affine minimal surfaces, and the subclass of surfaces with co-planar discrete harmonic co-normals, which we call discrete improper affine spheres. Within this classes, we show how to solve discrete Cauchy problems analogous to the Cauchy problems for smooth analytic improper affine spheres and smooth analytic affine minimal surfaces.

AMSclassification:primary 39A12; secondary 53A15, 52C99.

Keywords:discrete differential geometry, discrete affine minimal surfaces, discrete conjugate nets, PQ meshes.