Symplectic fillings of unit cotangent bundles of spheres and applications

We prove the uniqueness, up to diffeomorphism, of symplectically aspherical fillings of the unit cotangent bundle of the $3$-sphere S³ under a certain topological assumption, which Stein fillings automatically satisfy. In the course of the proof, we show that any symplectically aspherical filling of the unit cotangent bundle of the n-sphere Sⁿ (n ≥ 3) is simply-connected. As applications, we first show the non-existence of exact symplectic cobordisms between some $5$-dimensional Brieskorn manifolds. We also determine the diffeomorphism types of closed symplectic $6$-manifolds with certain codimension $2$ symplectic submanifolds.