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

Abstract

We address the following conjecture about the existence of common zeros for commuting vector fields in dimension 3: if are two commuting vector fields on a 3-manifold , and is a relatively compact open such that does not vanish on the boundary of and has a non-vanishing Poincaré–Hopf index in , then and have a common zero inside . We prove this conjecture when and are of class and every periodic orbit of along which and are collinear is partially hyperbolic. We also prove the conjecture, still in the setting, assuming that the flow leaves invariant a transverse plane field. These results shed new light on the case of the conjecture. This paper relies on colour figures. Some references to colour may not be meaningful in the printed version, and we refer the reader to the online version which includes the colour figures.