Regularity of Einstein 5-manifolds via 4-dimensional gap theorems

We refine the regularity of noncollapsed limits of Einstein 5-manifolds. In particular, we prove uniqueness of tangent cones along the full top stratum of singular set and show that the entire singular set is contained in a countable union of bi-Lipschitz curves and points. Moreover, we establish that thesingular curve carries a real-analytic Einstein orbifold structure and is a geodesic in the limit space. The proofs rely on new 4-dimensional gap theorems for spherical and hyperbolic Einstein orbifolds. This is joint work with Tristan Ozuch.