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

We address the following conjecture about the existence of common zeros for commuting vector fields in dimension 3: ifX,Yare twoC1commuting vector fields on a 3-manifoldM, andUis a relatively compact open such thatXdoes not vanish on the boundary ofUand has a non-vanishing Poincare-Hopf index inU, thenXandYhave a common zero insideU. We prove this conjecture whenXandYare of classC3and every periodic orbit ofYalong whichXandYare collinear is partially hyperbolic. We also prove the conjecture, still in theC3setting, assuming that the flowYleaves invariant a transverse plane field. These results shed new light on theC3case 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.