Curvature and injectivity radius estimates for Einstein 4-manifolds

Let M 4 M^4 denote an Einstein 4 4 -manifold with Einstein constant, λ λ , normalized to satisfy λ ∈ { − 3 , 0 , 3 } λ ∈ \{-3,0,3\} . For B r ( p ) ⊂ M 4 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 B1/2r , under the assumption that the L 2 L_2 -norm of the curvature on B r ( p ) B_r(p) is less than a small positive constant, which is independent of M 4 M^4 , and which in particular, does not depend on a lower bound on the volume of B r ( p ) B_r(p) . In case λ = − 3 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,