Curvature and injectivity radius estimates for Einstein 4-manifolds

Cheeger, Jeff¹ ; Tian, Gang²

¹ Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, New York 10012
² Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 and Department of Mathematics, Princeton University, Princeton, New Jersey, 08544

Journal of the American Mathematical Society, Tome 19 (2006) no. 2, pp. 487-525

Résumé

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, $-1\leq K_M0$. 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 $M^4$.

Détail

DOI : 10.1090/S0894-0347-05-00511-4

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