Cheeger–Colding–Tian Theory for Conic Kähler–Einstein Metrics


In this paper we extend the Cheeger–Colding–Tian theory to the conic Kahler–Einstein metrics. In general, there are no smooth approximations of a family of conic Kahler–Einstein metrics with Ricci curvature uniformly bounded from below. So we have to deal with the technical issues to extend the original arguments.

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  1. 1.

    Ambrosio, L., Gigli, N., Savar, G.: Metric measure spaces with Riemannian Ricci curvature bounded from below. Duke Math. J. 163(7), 1405–1490 (2014)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Bamler, R.: Structure theory of singular spaces, preprint. arXiv:1603.05236

  3. 3.

    Bo, H., Kell, M., Xia, C.: Harmonic functions on metric measure spaces, preprint. arXiv:1308.3607

  4. 4.

    Cheeger, J.: Degeneration of Riemannian Metrics Under Ricci Curvature Bounds. Scuola Normale Superiore, Pisa (2001)

    Google Scholar 

  5. 5.

    Cheeger, J.: Integral bounds on curvature, elliptic estimates and rectifiability of singular sets. Geom. Funct. Anal. 13(1), 20–72 (2003)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Cheeger, J., Colding, T.: Lower bounds on Ricci curvature and almost rigidity of warped product. Ann. Math. 144(1), 189–237 (1996)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Cheeger, J., Colding, T.: On the strcuture of spces with Ricci curvature bouned below: I. J. Differ. Geom. 46(3), 406–480 (1997)

    Article  Google Scholar 

  8. 8.

    Cheeger, J., Colding, T.: On the strcuture of spces with Ricci curvature bouned below: II. J. Differ. Geom. 54(1), 13–35 (2000)

    Article  Google Scholar 

  9. 9.

    Cheeger, J., Colding, T., Tian, G.: On the singularities of spaces with bounded Ricci curvature. Geom. Funct. Anal. 12(5), 873–914 (2002)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Colding, T.: Ricci curvature and the volume convergence. Ann. Math. 145(3), 477–504 (1997)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Dai, X.Z., Wei, G.F., Zhang, Z.L.: Neumann isoperimetric constant estimate for convex domains. arXiv:1612.05843

  12. 12.

    Datar, V.: On convexity of the regular set of conical Kähler–Einstein metrics. Math. Res. Lett. 23(1), 105–126 (2016)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Gigli, N.: The splitting theorem in non-smooth context, preprint. arXiv:1302.5555

  14. 14.

    Gigli, N., Philippis, G.: From volume cone to metric cone in the non smooth setting, preprint. arXiv:1512.03113

  15. 15.

    Li, C., Tian, G., Wang, F.: On Yau–Tian–Donaldson conjecture for singular Fano varieties, preprint. arXiv:1711.09530

  16. 16.

    Philippis, G., Gigli, N.: Non-collapsed spaces with Ricci curvature bounded from below, preprint. arXiv:1708.02060

  17. 17.

    Tian, G.: K-stability and Kähler–Einstein metrics. Commun. Pure Appl. Math. 68(7), 1085–1156 (2015)

    Article  Google Scholar 

  18. 18.

    Tian, G., Wang, F.: On the existence of conic Kähler–Einsten metrics, preprint

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Correspondence to Feng Wang.

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Gang Tian: Partially supported by NSFC Grants 11331001. Feng Wang: Partially supported by NSFC Grants 11501501 and the Fundamental Research Funds for the Central Universities 2018QNA3001.

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Tian, G., Wang, F. Cheeger–Colding–Tian Theory for Conic Kähler–Einstein Metrics. J Geom Anal 31, 1471–1509 (2021).

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  • Cheeger–Colding–Tian theory
  • Conic Kahler–Einstein metrics

Mathematics Subject Classification

  • 53C25