Milnor open books and Milnor fillable contact 3-manifolds

We say that an oriented contact manifold ( M , ξ ) is Milnor fillable if it is contactomorphic to the contact boundary of an isolated complex-analytic singularity ( X , x ) . In this article we prove that any three-dimensional oriented manifold admits at most one Milnor fillable contact structure up to contactomorphism. The proof is based on Milnor open books: we associate an open book decomposition of M with any holomorphic function f : ( X , x ) → ( C , 0 ) , with isolated singularity at x and we verify that all these open books carry the contact structure ξ of ( M , ξ ) —generalizing results of Milnor and Giroux.