Discrete Affine Minimal Surfaces with Indefinite Metric

TL;DR

This paper introduces a method to construct discrete affine minimal surfaces with indefinite metrics, extending smooth surface theory into a discrete setting using affine area functionals and discrete differential geometry tools.

Contribution

It presents a novel constructive process for discrete affine minimal surfaces, including definitions of discrete geometric objects and compatibility conditions, inspired by smooth surface theory.

Findings

Constructed a large class of discrete affine minimal surfaces

Defined discrete analogs of differential geometric objects

Established compatibility equations for these discrete surfaces

Abstract

Inspired by the Weierstrass representation of smooth affine minimal surfaces with indefinite metric, we propose a constructive process producing a large class of discrete surfaces that we call discrete affine minimal surfaces. We show that they are critical points of an affine area functional defined on the space of quadrangular discrete surfaces. The construction makes use of asymptotic coordinates and allows defining the discrete analogs of some differential geometric objects, such as the normal and co normal vector fields, the cubic form and the compatibility equations.