Manifolds with 1/4-pinched curvature are space forms

Brendle, Simon; Schoen, Richard

doi:10.1090/S0894-0347-08-00613-9

Manifolds with $1/4$-pinched curvature are space forms
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J. Amer. Math. Soc. 22 (2009), 287-307 Request permission

Abstract:

Let $(M,g_0)$ be a compact Riemannian manifold with pointwise $1/4$-pinched sectional curvatures. We show that the Ricci flow deforms $g_0$ to a constant curvature metric. The proof uses the fact, also established in this paper, that positive isotropic curvature is preserved by the Ricci flow in all dimensions. We also rely on earlier work of Hamilton and of Böhm and Wilking.

References

Four-manifolds with $1/4$-pinched flag curvatures,

M. Berger, Les variétés Riemanniennes $(1/4)$-pincées, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3) 14 (1960), 161–170 (French). MR 140054

Manifolds with positive curvature operator are space forms,

Simon Brendle and Richard M. Schoen, Classification of manifolds with weakly $1/4$-pinched curvatures, Acta Math. 200 (2008), no. 1, 1–13. MR 2386107, DOI 10.1007/s11511-008-0022-7
Bing-Long Chen and Xi-Ping Zhu, Ricci flow with surgery on four-manifolds with positive isotropic curvature, J. Differential Geom. 74 (2006), no. 2, 177–264. MR 2258799
Haiwen Chen, Pointwise $\frac 14$-pinched $4$-manifolds, Ann. Global Anal. Geom. 9 (1991), no. 2, 161–176. MR 1136125, DOI 10.1007/BF00776854
Ailana M. Fraser, Fundamental groups of manifolds with positive isotropic curvature, Ann. of Math. (2) 158 (2003), no. 1, 345–354. MR 1999925, DOI 10.4007/annals.2003.158.345
Detlef Gromoll, Differenzierbare Strukturen und Metriken positiver Krümmung auf Sphären, Math. Ann. 164 (1966), 353–371 (German). MR 196754, DOI 10.1007/BF01350046
Karsten Grove, Hermann Karcher, and Ernst A. Ruh, Group actions and curvature, Invent. Math. 23 (1974), 31–48. MR 385750, DOI 10.1007/BF01405201
Karsten Grove, Hermann Karcher, and Ernst A. Ruh, Jacobi fields and Finsler metrics on compact Lie groups with an application to differentiable pinching problems, Math. Ann. 211 (1974), 7–21. MR 355917, DOI 10.1007/BF01344138
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
Richard S. Hamilton, The formation of singularities in the Ricci flow, Surveys in differential geometry, Vol. II (Cambridge, MA, 1993) Int. Press, Cambridge, MA, 1995, pp. 7–136. MR 1375255
Richard S. Hamilton, Four-manifolds with positive isotropic curvature, Comm. Anal. Geom. 5 (1997), no. 1, 1–92. MR 1456308, DOI 10.4310/CAG.1997.v5.n1.a1
Gerhard Huisken, Ricci deformation of the metric on a Riemannian manifold, J. Differential Geom. 21 (1985), no. 1, 47–62. MR 806701
Hans-Christoph Im Hof and Ernst A. Ruh, An equivariant pinching theorem, Comment. Math. Helv. 50 (1975), no. 3, 389–401. MR 385751, DOI 10.1007/BF02565758
Hermann Karcher, A short proof of Berger’s curvature tensor estimates, Proc. Amer. Math. Soc. 26 (1970), 642–644. MR 270304, DOI 10.1090/S0002-9939-1970-0270304-5
Wilhelm Klingenberg, Über Riemannsche Mannigfaltigkeiten mit positiver Krümmung, Comment. Math. Helv. 35 (1961), 47–54 (German). MR 139120, DOI 10.1007/BF02567004
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
Mario J. Micallef and John Douglas Moore, Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes, Ann. of Math. (2) 127 (1988), no. 1, 199–227. MR 924677, DOI 10.2307/1971420
Mario J. Micallef and McKenzie Y. Wang, Metrics with nonnegative isotropic curvature, Duke Math. J. 72 (1993), no. 3, 649–672. MR 1253619, DOI 10.1215/S0012-7094-93-07224-9

Invariant curvature cones and the Ricci flow,

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

The entropy formula for the Ricci flow and its geometric applications,

Ricci flow with surgery on three-manifolds,

H. E. Rauch, A contribution to differential geometry in the large, Ann. of Math. (2) 54 (1951), 38–55. MR 42765, DOI 10.2307/1969309
Ernst A. Ruh, Krümmung und differenzierbare Struktur auf Sphären. II, Math. Ann. 205 (1973), 113–129 (German). MR 370429, DOI 10.1007/BF01350841
Ernst A. Ruh, Riemannian manifolds with bounded curvature ratios, J. Differential Geometry 17 (1982), no. 4, 643–653 (1983). MR 683169
M. Sugimoto and K. Shiohama, On the differentiable pinching problem, Math. Ann. 195 (1971), 1–16. Improved by H Karcher. MR 290291, DOI 10.1007/BF02059412