Vector fields and genus in dimension 3

TL;DR

This paper derives a formula linking the Euler characteristic of transverse surfaces in 3D flows to boundary data, and shows genus relates to helicity as an asymptotic invariant.

Contribution

It introduces a new formula for Euler characteristic in 3D flows and connects genus to helicity as an asymptotic invariant.

Findings

Euler characteristic formula based on boundary data

Genus is proportional to helicity in ergodic flows

Examples illustrating low genus surfaces

Abstract

Given a flow on a 3-dimensional integral homology sphere, we give a formula for the Euler characteristic of its transverse surfaces, in terms of boundary data only. We illustrate the formula with several examples, in particular with surfaces of low genus. As an application, we show that for a right-handed flow with an ergodic invariant measure, the genus is an asymptotic invariant of order 2 proportional to the helicity.