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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Curvature and injectivity radius estimates for Einstein 4-manifolds
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by Jeff Cheeger and Gang Tian PDF
J. Amer. Math. Soc. 19 (2006), 487-525 Request permission

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
  • 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