Milnor open books and Milnor fillable contact 3-manifolds

We say that an oriented contact manifold (M,ξ)(M,ξ) is Milnor fillable   if it is contactomorphic to the contact boundary of an isolated complex-analytic singularity (X,x)(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)f:(X,x)→(C,0), with isolated singularity at x   and we verify that all these open books carry the contact structure ξξ of (M,ξ)(M,ξ)—generalizing results of Milnor and Giroux.