Homoclinic tangencies and singular hyperbolicity for three-dimensional vector fields

Abstract: We prove that any vector field on a three-dimensional compact manifold can be approximated in the C1-topology by one which is singular hyperbolic or by one which exhibits a homoclinic tangency associated to a regular hyperbolic periodic orbit. This answers a conjecture by Palis. During the proof we obtain several other results with independent interest: a compactification of the rescaled sectional Poincar\'e flow and a generalization of Ma~n\'e-Pujals-Sambarino theorem for three-dimensional C2 vector fields with singularities.