Characterizations of lattice surfaces

TL;DR

This paper characterizes lattice surfaces by establishing equivalences with geometric properties, showing finiteness of classes with bounded triangle areas, and providing explicit bounds, thereby deepening understanding of their structure.

Contribution

It demonstrates the equivalence between the lattice property and a positive lower bound for affine triangle areas, and provides explicit bounds on classes with such bounds.

Findings

Lattice property is equivalent to a positive lower bound for affine triangle areas.

The set of lattice surfaces with a fixed lower bound is finite.

Explicit bounds on the number of such lattice surfaces are obtained.

Abstract

We answer a question of Vorobets by showing that the lattice property for flat surfaces is equivalent to the existence of a positive lower bound for the areas of affine triangles. We show that the set of affine equivalence classes of lattice surfaces with a fixed positive lower bound for the areas of triangles is finite and we obtain explicit bounds on its cardinality. We deduce several other characterizations of the lattice property.