## Curvature and injectivity radius estimates for Einstein 4-manifolds

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- by Jeff Cheeger and Gang Tian
- J. Amer. Math. Soc.
**19**(2006), 487-525 - DOI: https://doi.org/10.1090/S0894-0347-05-00511-4
- Published electronically: December 2, 2005
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## 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 $M^4$.## References

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## Bibliographic Information

**Jeff Cheeger**- Affiliation: Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, New York 10012
- MR Author ID: 47805
- Email: cheeger@cims.nyu.edu
**Gang Tian**- Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 and Department of Mathematics, Princeton University, Princeton, New Jersey, 08544
- MR Author ID: 220655
- Email: tian@math.princeton.edu
- Received by editor(s): December 2, 2004
- Published electronically: December 2, 2005
- Additional Notes: The first author was partially supported by NSF Grant DMS 0104128

The second author was partially supported by NSF Grant DMS 0302744 - © Copyright 2005 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**19**(2006), 487-525 - MSC (2000): Primary 53Cxx
- DOI: https://doi.org/10.1090/S0894-0347-05-00511-4
- MathSciNet review: 2188134