Curvature and injectivity radius estimates for Einstein 4-manifolds

Abstract

Let $M^4$ denote an Einstein $4$-manifold with Einstein constant, $\lambda$, normalized to satisfy $\lambda \in \{-3,0,3\}$. For $B_r(p)\subset M^4$, a metric ball, we prove a uniform estimate for the pointwise norm of the curvature tensor on $B_{\frac {1}{2}r}$, 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 $\lambda =-3$, we prove a lower injectivity radius bound analogous to that which occurs in the theorem of Margulis, for compact manifolds with negative sectional curvature, $-1\leq K_M<0$. 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 $M^4$.