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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Ricci flow, Einstein metrics and space forms
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by Rugang Ye PDF
Trans. Amer. Math. Soc. 338 (1993), 871-896 Request permission

Abstract:

The main results in this paper are: (1) Ricci pinched stable Riemannian metrics can be deformed to Einstein metrics through the Ricci flow of R. Hamilton; (2) (suitably) negatively pinched Riemannian manifolds can be deformed to hyperbolic space forms through Ricci flow; and (3) ${L^2}$-pinched Riemannian manifolds can be deformed to space forms through Ricci flow.
References
  • Arthur L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 10, Springer-Verlag, Berlin, 1987. MR 867684, DOI 10.1007/978-3-540-74311-8
  • Dennis M. DeTurck and Jerry L. Kazdan, Some regularity theorems in Riemannian geometry, Ann. Sci. École Norm. Sup. (4) 14 (1981), no. 3, 249–260. MR 644518
  • F. T. Farrell and L. E. Jones, Negatively curved manifolds with exotic smooth structures, J. Amer. Math. Soc. 2 (1989), no. 4, 899–908. MR 1002632, DOI 10.1090/S0894-0347-1989-1002632-2
  • Sylvestre Gallot, Inégalités isopérimétriques, courbure de Ricci et invariants géométriques. I, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), no. 7, 333–336 (French, with English summary). MR 697966
  • L. Gao, Convergence of Riemannian manifolds, Ricci pinching and ${L^{N/2}}$-curvature pinching, preprint.
  • M. Gromov, Manifolds of negative curvature, J. Differential Geometry 13 (1978), no. 2, 223–230. MR 540941
  • M. Gromov and W. Thurston, Pinching constants for hyperbolic manifolds, Invent. Math. 89 (1987), no. 1, 1–12. MR 892185, DOI 10.1007/BF01404671
  • Richard S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geometry 17 (1982), no. 2, 255–306. MR 664497
  • Richard S. Hamilton, Four-manifolds with positive curvature operator, J. Differential Geom. 24 (1986), no. 2, 153–179. MR 862046
  • —, Lectures on heat equations in geometry, Lecture Notes, Hawaii, 1989.
  • Gerhard Huisken, Ricci deformation of the metric on a Riemannian manifold, J. Differential Geom. 21 (1985), no. 1, 47–62. MR 806701
  • Norihito Koiso, Nondeformability of Einstein metrics, Osaka Math. J. 15 (1978), no. 2, 419–433. MR 504300
  • Norihito Koiso, On the second derivative of the total scalar curvature, Osaka Math. J. 16 (1979), no. 2, 413–421. MR 539596
  • Norihito Koiso, Rigidity and stability of Einstein metrics—the case of compact symmetric spaces, Osaka Math. J. 17 (1980), no. 1, 51–73. MR 558319
  • Peter Li, On the Sobolev constant and the $p$-spectrum of a compact Riemannian manifold, Ann. Sci. École Norm. Sup. (4) 13 (1980), no. 4, 451–468. MR 608289
  • Christophe Margerin, Pointwise pinched manifolds are space forms, Geometric measure theory and the calculus of variations (Arcata, Calif., 1984) Proc. Sympos. Pure Math., vol. 44, Amer. Math. Soc., Providence, RI, 1986, pp. 307–328. MR 840282, DOI 10.1090/pspum/044/840282
  • Maung Min-Oo, Almost Einstein manifolds of negative Ricci curvature, J. Differential Geom. 32 (1990), no. 2, 457–472. MR 1072914
  • M. Min-Oo and E. A. Ruh, ${L^2}$-curvature pinching, preprint.
  • Jürgen Moser, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math. 17 (1964), 101–134. MR 159139, DOI 10.1002/cpa.3160170106
  • Seiki Nishikawa, Deformation of Riemannian metrics and manifolds with bounded curvature ratios, Geometric measure theory and the calculus of variations (Arcata, Calif., 1984) Proc. Sympos. Pure Math., vol. 44, Amer. Math. Soc., Providence, RI, 1986, pp. 343–352. MR 840284, DOI 10.1090/pspum/044/840284
  • Stefan Peters, Convergence of Riemannian manifolds, Compositio Math. 62 (1987), no. 1, 3–16. MR 892147
  • Ernst A. Ruh, Riemannian manifolds with bounded curvature ratios, J. Differential Geometry 17 (1982), no. 4, 643–653 (1983). MR 683169
  • Richard M. Schoen, Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, Topics in calculus of variations (Montecatini Terme, 1987) Lecture Notes in Math., vol. 1365, Springer, Berlin, 1989, pp. 120–154. MR 994021, DOI 10.1007/BFb0089180
  • Wan-Xiong Shi, Ricci deformation of the metric on complete noncompact Riemannian manifolds, J. Differential Geom. 30 (1989), no. 2, 303–394. MR 1010165
  • D. Yang, ${L^p}$ pinching and compactness theorems for compact Riemannian manifolds, preprint.
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Additional Information
  • © Copyright 1993 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 338 (1993), 871-896
  • MSC: Primary 58E11; Secondary 53C25, 58G11
  • DOI: https://doi.org/10.1090/S0002-9947-1993-1108615-3
  • MathSciNet review: 1108615