Four-manifolds with positive isotropic curvature

Abstract

We give a survey on 4-dimensional manifolds with positive isotropic curvature. We will introduce the work of B. L. Chen, S. H. Tang and X. P. Zhu on a complete classification theorem on compact four-manifolds with positive isotropic curvature (PIC). Then we review an application of the classification theorem, which is from Chen and Zhu’s work. Finally, we discuss our recent result on the path-connectedness of the moduli spaces of Riemannian metrics with positive isotropic curvature.

Keywords

Four-manifolds / positive isotropic curvature (PIC) / Ricci flow

References

[1] Brendle S, Schoen R. Manifolds with 1/4-pinched curvature are space forms. J Amer Math Soc, 2009, 22: 287–307

[2] Brendle S, Schoen R. Sphere theorems in geometry. Surv Differ Geom, 2009, 13: 49–84

[3] Carr R. Construction of manifolds of positive scalar curvature. Trans Amer Math Soc, 1988, 307: 63–74

[4] Cerf J. Sur les difféomorphismes de la sph`ere de dimension trois (Γ4= 0).Lecture Notes in Math, Vol 53. Berlin: Springer-Verlag, 1968

[5] Chang S-Y A, Gursky M J, Yang P. A conformally invariant sphere theorem in four dimensions. Publ Math Inst Hautes ´Etudes Sci, 2003, 98: 105–143

[6] Chen B L, Huang X T. Path-connectedness of the moduli spaces of metrics with positive isotropic curvature on four-manifolds. Math Ann (to appear),

[7] Chen B L, Tang S H, Zhu X P. Complete classification of compact four manifolds with positive isotropic curvature. J Differential Geom, 2012, 91: 41–80

[8] Chen B L, Zhu X P. Ricci flow with surgery on four-manifolds with positive isotropic curvature. J Differential Geom, 2006, 74: 177–264

[9] Chen B L, Zhu X P. A conformally invariant classification theorem in four dimensions. Comm Anal Geom, 2014, 22: 811–831

[10] DeTurck D M. Deforming metrics in the direction of their Ricci tensors. J Differential Geom, 1983, 18: 157–162

[11] Farrell F T, Ontaneda P. On the topology of the space of negatively curved metrics. J Differential Geom, 2010, 86: 273–301

[12] Fraser A M. Fundamental groups of manifolds with positive isotropic curvature. Ann of Math, (2), 2003, 158: 345–354

[13] Gromov M. Positive Curvature, Macroscopic Dimension, Spectral Gaps and Higher Signatures. Progr Math, Vol 132. Basel: Birkhäuser, 1996

[14] Gromov M, Lawson H B Jr. Positive scalar curvature and the Dirac operator on complete Riemannian manifolds. Publ Math Inst Hautes Études Sci, 1983, 58: 83–196

[15] Gursky M J, LeBrun C. Yamabe invariants and Spinc structures. Geom Funct Anal, 1998, 8: 965–977

[16] Hamilton R. Three-manifolds with positive Ricci curvature. J Differential Geom, 1982, 17: 255–306

[17] Hamilton R. Four-manifolds with positive curvature operator. J Differential Geom, 1986, 24: 153–179

[18] Hamilton R. Four-manifolds with positive isotropic curvature. Comm Anal Geom, 1997, 5: 1–92

[19] Hamilton R. Three-orbifolds with positive Ricci curvature. In: Collected Papers on Ricci Flow. Ser Geom Topol, Vol 37. Somerville: Int Press, 2003, 521–524

[20] Hitchin N. Harmonic spinors. Adv Math, 1974, 14: 1–55

[21] Kapovitch V, Petrunin A, Tuschmann W. Non-negative pinching, moduli spaces and bundles with infinitely many souls. J Differential Geom, 2005, 71: 365–383

[22] Kreck M, Stolz S. Nonconnected moduli spaces of positive sectional curvature metrics. J Amer Math Soc, 1993, 6: 825–850

[23] Lohkamp J. The space of negative scalar curvature metrics. Invent Math, 1992, 110: 403–407

[24] Marques F C. Deforming three-manifolds with positive scalar curvature. Ann of Math (2), 2012, 176: 815–863

[25] McCullough D. Isometries of elliptic 3-manifolds. J Lond Math Soc (2), 2002, 65: 167–182

[26] Micallef M J, Moore J D. Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes. Ann of Math (2), 1988, 127: 199–227

[27] Micallef M J, Wang M. Metrics with nonnegative isotropic curvature. Duke Math J, 1993, 72: 649–672

[28] Nguyen H T. Isotropic curvature and the Ricci flow. Int Math Res Not IMRN, 2010, 2010(3): 536–558

[29] Rosenberg J. Manifolds of positive scalar curvature: a progress report. In: Surv Differ Geom, Vol 11. Somerville: Int Press, 2007, 259–294

[30] Rosenberg J, Stolz S. Metrics of positive scalar curvature and connections with surgery. In: Surveys on Surgery Theory: Papers Dedicated to C. T. C. Wall, Vol 2. Ann of Math Stud, Vol 149. Princeton: Princeton Univ Press, 2001, 353–386

[31] Ruberman D. Positive scalar curvature, diffeomorphisms and the Seiberg-Witten invariants. Geom Topol, 2001, 5: 895–924

[32] Schoen R. Conformal deformation of a Riemannian metric to constant scalar curvature. J Differential Geom, 1984, 20: 479–495

[33] Schoen R. Minimal submanifolds in higher codimension. Mat Contemp, 2006, 30: 169–199

[34] Scott P. The geometries of 3-manifolds. Bull Lond Math Soc, 1983, 15: 401–487

[35] Ue M. Geometric 4-manifolds in the sense of Thurston and Seifert 4-manifolds II. J Math Soc Japan, 1991, 43: 149–183

[36] Weyl H. Über die Bestimmung einer geschlossenen konvexen Fläche durch ihr Linienelement. Vierteljahrsschr Naturforsch Ges Zür, 1916, 61: 40–72

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