Abstract
Under the assumption of the uniform local Sobolev inequality, it is proved that Riemannian metrics with an absolute Ricci curvature bound and a small Riemannian curvature integral bound can be smoothed to having a sectional curvature bound. This partly extends previous a priori estimates of Li (J Geom Anal 17:495–511, 2007; Adv Math 223:1924–1957, 2010).
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References
Anderson M.: The L 2 structure of moduli spaces of Einstein metrics on 4-manifolds. Geom. Funct. Anal. 2, 29–89 (1992)
Bando S.: Real analyticity of solutions of Hamilton’s equation. Math. Z. 195, 93–97 (1987)
Bemelmans J., Min-Oo M., Ruh E.: Smoothing Riemannian metrics. Math. Z. 188, 69–74 (1984)
Cheeger, J.: Critical points of distance functions and applications to geometry. Lecture Notes in Math., vol. 1504, Springer Verlag, pp. 1–38 (1991)
Cheeger J., Tian G.: Curvature and injectivity radius estimates for Einstein 4-manifolds. J. Am. Math. Soc. 19, 487–525 (2006)
Dai X., Wei G., Ye R.: Smoothing Riemannian metrics with Ricci curvature bounds. Manuscripta Math. 90, 49–61 (1996)
Dai, X., Wei, G., Ye, R.: Smoothing Riemannian metrics with Ricci curvature bounds. arXiv, dg-ga/9411014, 1994
DeTurk D.: Deforming metrics in the direction of the direction of their Ricci tensors. J. Diff. Geom. 18, 157–162 (1983)
Gromov M.: Almost flat manifolds. J. Diff. Deom. 13, 231–241 (1978)
Hamilton R.: Three-manifolds with positive Ricci curvature. J. Diff. Geom. 17, 255–306 (1982)
Li Y.: Local volume estimate for manifolds with L 2-bounded curvature. J. Geom. Anal. 17, 495–511 (2007)
Li Y.: Smoothing Riemannian metrics with bounded Ricci curvatures in dimension four. Adv. Math. 223, 1924–1957 (2010)
Li, Y.: Smoothing Riemannian metrics with bounded Ricci curvatures in dimension four, II. arXiv, 0911.3104v1, 2009
Shi W.: Deforming the metric on complete Riemannian manifolds. J. Diff. Geom. 30, 223–301 (1989)
Tian G., Viaclovsky J.: Bach-flat asymptotically locally Euclidean metrics. Invent. Math. 160, 357–415 (2005)
Xu, G.: Short-time existence of the Ricci flow on noncompact Riemannian manifolds. arXiv, 0907.5604v1, 2009
Yang, D.: L p pinching and compactness theorems for compact Riemannian manifolds. Sémminaire de théorie spectrale et géométrie, Chambéry-Grenoble, 1987–1988, pp. 81–89
Yang D.: Convergence of Riemannian manifolds with integral bounds on curvature I. Ann. Sci. Ecole Norm. Sup. 25, 77–105 (1992)
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Yang, Y. Smoothing metrics on closed Riemannian manifolds through the Ricci flow. Ann Glob Anal Geom 40, 411–425 (2011). https://doi.org/10.1007/s10455-011-9262-4
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DOI: https://doi.org/10.1007/s10455-011-9262-4