Existence of common zeros for commuting vector fields on $3$-manifolds II. Solving global difficulties

TL;DR

This paper proves a conjecture about the existence of common zeros for commuting vector fields on 3-manifolds under certain hyperbolicity and invariance conditions, advancing understanding in differential topology.

Contribution

It establishes the conjecture for $C^3$ vector fields when periodic orbits are partially hyperbolic or when the flow preserves a transverse plane field.

Findings

Proved the conjecture for $C^3$ vector fields with partially hyperbolic periodic orbits.

Confirmed the conjecture when the flow preserves a transverse plane field.

Enhanced understanding of common zeros in the context of commuting vector fields on 3-manifolds.

Abstract

We address the following conjecture about the existence of common zeros for commuting vector fields in dimension three: if $X,Y$ are two $C^1$ commuting vector fields on a $3$-manifold $M$, and $U$ is a relatively compact open such that $X$ does not vanish on the boundary of $U$ and has a non vanishing Poincar\'e-Hopf index in $U$, then $X$ and $Y$ have a common zero inside $U$. We prove this conjecture when $X$ and $Y$ are of class $C^3$ and every periodic orbit of $Y$ along which $X$ and $Y$ are collinear is partially hyperbolic. We also prove the conjecture, still in the $C^3$ setting, assuming that the flow $Y$ leaves invariant a transverse plane field. These results shed new light on the $C^3$ case of the conjecture.