Discrete projective minimal surfaces

TL;DR

This paper introduces a discretisation scheme for classical projective minimal surfaces, creating discrete models of geometric notions and classifying these surfaces based on their envelopes, leading to new discrete surface types.

Contribution

It provides the first systematic discretisation of projective minimal surfaces, including discrete analogues of classical surfaces and a classification scheme based on envelope properties.

Findings

Discrete models of Lie quadrics and envelopes are introduced.

Discrete projective minimal surfaces are classified by envelope cardinality.

New discrete surface classes such as discrete Q surfaces are defined.

Abstract

We propose a natural discretisation scheme for classical projective minimal surfaces. We follow the classical geometric characterisation and classification of projective minimal surfaces and introduce at each step canonical discrete models of the associated geometric notions and objects. Thus, we introduce discrete analogues of classical Lie quadrics and their envelopes and classify discrete projective minimal surfaces according to the cardinality of the class of envelopes. This leads to discrete versions of Godeaux-Rozet, Demoulin and Tzitzeica surfaces. The latter class of surfaces requires the introduction of certain discrete line congruences which may also be employed in the classification of discrete projective minimal surfaces. The classification scheme is based on the notion of discrete surfaces which are in asymptotic correspondence. In this context, we set down a discrete analogue of a classical theorem which states that an envelope (of the Lie quadrics) of a surface is in asymptotic correspondence with the surface if and only if the surface is either projective minimal or a Q surface. Accordingly, we present a geometric definition of discrete Q surfaces and their relatives, namely discrete counterparts of classical semi-Q, complex, doubly Q and doubly complex surfaces.