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arXiv · 1710.06743

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

Sébastien Alvarez·Christian Bonatti·Bruno Santiago
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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.

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Sébastien Alvarez, Christian Bonatti, Bruno Santiago. 2017-10-18. Existence of common zeros for commuting vector fields on $3$-manifolds II. Solving global difficulties. https://doi.org/10.1112/plms.12342

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