On the stable norm of slit tori and the Farey sequence

TL;DR

This paper computes the stable norm on flat slit tori using Farey sequences, constructs complex surfaces with maximal faces in their stable norm unit ball, and estimates the growth of simple homology classes.

Contribution

It provides explicit stable norm calculations for slit tori, constructs new surfaces with maximal faces in the stable norm, and offers sub-quadratic growth estimates for homology classes.

Findings

Explicit stable norm computations for flat slit tori.

Construction of surfaces with maximal faces in the stable norm.

Sub-quadratic estimate for counting simple homology classes.

Abstract

Let (M, g) be a compact manifold endowed with a possibly singular Riemannian metric. The metric induces a norm on the homology of M , called the stable norm. We provide explicit computations of the stable norm of flat slit tori using the Farey sequence. We then glue several slit tori together to produce half-translation surfaces whose unit ball of the stable norm has faces of maximal dimension. Furthermore, we give a sub-quadratic estimate for the asymptotic counting of simple homology classes on these surfaces.