Curvature and injectivity radius estimates for Einstein 4-manifolds

Abstract

Let M 4 denote an Einstein 4 -manifold with Einstein constant, λ , normalized to satisfy λ ∈ { − 3 , 0 , 3 } . For B r ( p ) ⊂ M 4 , a metric ball, we prove a uniform estimate for the pointwise norm of the curvature tensor on B 1 2 r , under the assumption that the L 2 -norm of the curvature on B r ( p ) is less than a small positive constant, which is independent of M 4 , and which in particular, does not depend on a lower bound on the volume of B r ( p ) . In case λ = − 3 , we prove a lower injectivity radius bound analogous to that which occurs in the theorem of Margulis, for compact manifolds with negative sectional curvature, − 1 ≤ K M > 0 . These estimates provide key tools in the study of singularity formation for 4 -dimensional Einstein metrics. As one application among others, we give a natural compactification of the moduli space of Einstein metrics with negative Einstein constant on a given M 4 .