Discrete Asymptotic Nets with Constant Affine Mean Curvature

TL;DR

This paper introduces a new class of discrete asymptotic nets with constant affine mean curvature, linking affine spheres and minimal nets, and explores their properties and transformations.

Contribution

It defines CAMC discrete asymptotic nets via compatible interpolating quadrics, extending classical smooth surface properties to the discrete setting.

Findings

CAMC nets include affine spheres and minimal asymptotic nets

Ruled discrete asymptotic nets are equivalent to ruled compatible interpolating quadrics

Discrete Demoulin transform properties are established for CAMC surfaces

Abstract

In this paper we define the class of constant affine mean curvature (CAMC) discrete asymptotic nets, which contains the well-known classes of affine spheres and affine minimal asymptotic nets. This class is defined by considering fields of compatible interpolating quadrics, i.e., quadrics that have common tangent planes at the edges of the net. We show that, for CAMC asymptotic nets, ruled discrete asymptotic nets is equivalent to ruled compatible interpolating quadrics. Moreover, we prove discrete counterparts of some known properties of the Demoulin transform of a smooth CAMC surface.