Abstract
We prove the weight part of Serre's conjecture in generic situations for forms of U (3) which are compact at infinity and split at places dividing p as conjectured by [Her09]. We also prove automorphy lifting theorems in dimension three. The key input is an explicit description of tamely potentially crystalline deformation rings with Hodge-Tate weights (2, 1, 0) for K/Qp unramified combined with patching techniques. Our results show that the (geometric) Breuil-Mézard conjectures hold for these deformation rings.