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Curvature and injectivity radius estimates for Einstein 4-manifolds

Authors: Jeff Cheeger and Gang Tian
Journal: J. Amer. Math. Soc. 19 (2006), 487-525
MSC (2000): Primary 53Cxx
Published electronically: December 2, 2005
MathSciNet review: 2188134
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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$.

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Additional Information

Jeff Cheeger
Affiliation: Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, New York 10012

Gang Tian
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 and Department of Mathematics, Princeton University, Princeton, New Jersey, 08544

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
Article copyright: © Copyright 2005 American Mathematical Society

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