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A continuity method to construct canonical metrics

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Abstract

We introduce a new continuity method which, although less natural than flows such as the Kähler–Ricci flow, has the advantage of preserving a lower bound on the Ricci curvature, hence allowing the application of comparison geometry techniques, such as Cheeger–Colding–Tian’s compactness theory.

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Notes

  1. Both \(C'\) and \( C''\) are uniform constants.

  2. One may also use the \(C^{2,\alpha }\)-estimate (cf. [2, 7]).

  3. Since \(\omega _t\) has Ricci curvature bounded from below, this is equivalent to that the diameter of \((M,\omega _t)\) is uniformly bounded.

  4. \(M_i\) may be of smaller dimension or even an empty set.

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Authors and Affiliations

  1. University of Illinois, Champaign, USA

    Gabriele La Nave

  2. Beijing University, Beijing, China

    Gang Tian

  3. Princeton University, Princeton, China

    Gang Tian

Authors
  1. Gabriele La Nave
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  2. Gang Tian
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Corresponding author

Correspondence to Gabriele La Nave.

Additional information

G. Tian was supported partially by grants from NSF and NSFC.

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La Nave, G., Tian, G. A continuity method to construct canonical metrics. Math. Ann. 365, 911–921 (2016). https://doi.org/10.1007/s00208-015-1255-3

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  • DOI: https://doi.org/10.1007/s00208-015-1255-3

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