Regularisation of Lie algebroids and applications

Abstract

We describe a procedure, called regularisation, that allows us to study geometric structures on Lie algebroids via foliated geometric structures on manifolds of higher dimension. This procedure applies to various classes of Lie algebroids; namely, those whose singularities are of bk, complex-log, or elliptic type, possibly with self-crossings.One of our main applications is a proof of the Weinstein conjecture for overtwisted bk-contact structures. This was proven in [23] using a certain technical hypothesis. Our approach avoids this assumption by reducing the proof to the foliated setting. As a by-product, we also prove the Weinstein conjecture for other Lie algebroids.