On a Conformal Gap and Finiteness Theorem for a Class of Four-Manifolds

Abstract

In this paper we develop a bubble tree structure for a degenerating class of Riemannian metrics satisfying some global conformal bounds on compact manifolds of dimension 4. Applying the bubble tree structure, we establish a gap theorem, a finiteness theorem for diffeomorphism type for this class, and make a comparison of the solutions of the σk equations on a degenerating family of Bach-flat metrics.