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Expansions of solutions to extremal metric type equations on blowups of cscK surfaces

Abstract

The aim of this article is to study expansions of solutions to an extremal metric type equation on the blowup of constant scalar curvature Kähler surfaces. This is related to finding constant scalar curvature Kähler (cscK) metrics on K-stable blowups of extremal Kähler surfaces (Székelyhidi in Duke Math J 161(8):1411–1453, 2012).

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References

  1. Arezzo, C., Lena, R., Mazzieri, L.: On the resolution of extremal and constant scalar curvature Kähler orbifolds. Int. Math. Res. Not. IMRN 21, 6415–6452 (2016)

    Article  MATH  Google Scholar 

  2. Arezzo, C., Pacard, F.: Blowing up and desingularizing constant scalar curvature Kähler manifolds. Acta Math. 196(2), 179–228 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  3. Arezzo, C., Pacard, F.: Blowing up Kähler manifolds with constant scalar curvature. II. Ann. of Math. 170(2), 685–738 (2009)

  4. Arezzo, C., Pacard, F., Singer, M.: Extremal metrics on blowups. Duke Math. J. 157(1), 1–51 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  5. Bartnik, R.: The mass of an asymptotically flat manifold. Comm. Pure Appl. Math. 39(5), 661–693 (1986)

    MathSciNet  Article  MATH  Google Scholar 

  6. Bochner, S., Martin, W.T.: Several Complex Variables. Princeton Mathematical Series, vol. 10. Princeton University Press, Princeton (1948)

    MATH  Google Scholar 

  7. Chen, X.X., Donaldson, S.K., Sun, S.: Kähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities. J. Amer. Math. Soc. 28(1), 183–197 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  8. Chen, X.X., Donaldson, S.K., Sun, S.: Kähler–Einstein metrics on Fano manifolds. II: Limits with cone angle less than \(2\pi \). J. Amer. Math. Soc. 28(1), 199–234 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  9. Chen, X.X., Donaldson, S.K., Sun, S.: Kähler–Einstein metrics on fano manifolds. iii: Limits as cone angle approaches \(2\pi \) and completion of the main proof. J. Amer. Math. Soc. 102(1), 235–278 (2015)

    Article  MATH  Google Scholar 

  10. Donaldson, S.K.: Scalar curvature and stability of toric varieties. J. Differential Geom. 62(2), 289–349 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  11. Gursky, M.J., Viaclovsky, J.A.: Critical metrics on connected sums of Einstein four-manifolds. Adv. Math. 292, 210–315 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  12. LeBrun, C.: Counter-examples to the generalized positive action conjecture. Comm. Math. Phys. 118(4), 591–596 (1988)

    MathSciNet  Article  MATH  Google Scholar 

  13. LeBrun, C., Simanca, S.R.: Extremal Kähler metrics and complex deformation theory. Geom. Funct. Anal. 4(3), 298–336 (1994)

    MathSciNet  Article  MATH  Google Scholar 

  14. Mahmoudi, F., Mazzeo, R., Pacard, F.: Constant mean curvature hypersurfaces condensing on a submanifold. Geom. Funct. Anal. 16(4), 924–958 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  15. Melrose, R.B.: The Atiyah-Patodi-Singer Index Theorem. Research Notes in Mathematics, vol. 4. A K Peters Ltd, Wellesley (1993)

    Book  MATH  Google Scholar 

  16. Pacard, F., Rivière, T.: Linear and nonlinear aspects of vortices. In: Progress in Nonlinear Differential Equations and their Applications. The Ginzburg–Landau model, vol. 39. Birkhäuser Boston, Inc., Boston (2000)

  17. Seyyedali, R., Székelyhidi, G.: Extremal metrics on blowups along submanifolds. arxiv:1610.06865 (2016)

  18. Simanca, S.R.: Kähler metrics of constant scalar curvature on bundles over \({ C}{\rm P}_{n-1}\). Math. Ann. 291(2), 239–246 (1991)

    MathSciNet  Article  Google Scholar 

  19. Székelyhidi, G.: On blowing up extremal Kähler manifolds. Duke Math. J. 161(8), 1411–1453 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  20. Székelyhidi, G.: An Introduction to Extremal Kähler Metrics. Graduate Studies in Mathematics, vol. 152. American Mathematical Society, Providence (2014)

    MATH  Google Scholar 

  21. Székelyhidi, G.: Blowing up extremal Kähler manifolds II. Invent. Math. 200(3), 925–977 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  22. Tian, G.: Kähler–Einstein metrics with positive scalar curvature. Invent. Math. 130(1), 1–37 (1997)

    MathSciNet  Article  MATH  Google Scholar 

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

    MathSciNet  Article  MATH  Google Scholar 

  24. Yau, S.-T.: Open problems in geometry. In: Differential Geometry: Partial Differential Equations on Manifolds (Los Angeles, CA, 1990), Proc. Sympos. Pure Math., vol. 54, pp. 1–28. American Mathematical Society, Providence (1993)

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Acknowledgements

I would like to thank Gábor Székelyhidi for introducing me to Conjecture 1 and for sharing many useful insights. I would also like to thank Claudio Arezzo, Richard Bamler, Rafe Mazzeo, Michael Singer, Jian Song and Xiaowei Wang for stimulating discussions.

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Correspondence to Ved V. Datar.

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Research supported by NSF RTG Grant DMS-1344991.

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Datar, V.V. Expansions of solutions to extremal metric type equations on blowups of cscK surfaces. Ann Glob Anal Geom 55, 215–241 (2019). https://doi.org/10.1007/s10455-018-9624-2

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  • DOI: https://doi.org/10.1007/s10455-018-9624-2

Keywords

  • Extremal metrics
  • Gluing constructions
  • Blowups