Kähler Metrics with Cone Singularities Along a Divisor

Chapter

Abstract

We develop some analytical foundations for the study of Kähler metrics with cone singularities in codimension one. The main result is an analogue of the Schauder theory in this setting. In the later parts of the paper we discuss connections with the existence problem for Kähler–Einstein metrics,in the positive case.

Keywords

Line Bundle Heat Kernel Cone Angle Cone Singularity Einstein Metrics 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    R. Berman, A thermodynamic formalism for Monge-Ampère equations, Moser-Trudinger inequalities and Kähler–Einstein metrics. arxiv 1011.3976Google Scholar
  2. 2.
    O. Biquard, Sur les fibrées parabolique sur une surface complexes. J. Lond. Math. Soc. 253(2) 302–316 (1996)Google Scholar
  3. 3.
    H.S. Carslaw, The Green’s function for a wedge of any angle and other problems in the conduction of heat. Proc. Lond. Math. Soc. 8, 365–374 (1910)Google Scholar
  4. 4.
    D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equations of Second Order (Springer, Berlin/New York, 1983)Google Scholar
  5. 5.
    C. Hodgson, S. Kerchoff, Rigidity of hyperbolic cone manifolds and hyperbolic Dehn surgery. J. Differ. Geom. 48, 1–59 (1998)Google Scholar
  6. 6.
    T. Jeffres, Uniqueness of Kähler–Einstein cone metrics. Publ. Mat. 44(2), 437–448 (2000)Google Scholar
  7. 7.
    T. Jeffres, Schwarz lemma for Kähler cone metrics. Int. Math. Res. Not. 2000(7), 371–382 (2000)Google Scholar
  8. 8.
    P. Kronheimer, T. Mrowka, Gauge theory for embedded surfaces, I. Topology 32(4), 773–826, (1993)Google Scholar
  9. 9.
    C. Li, Greatest lower bounds on Ricci curvature for toric Fano manifolds. arxiv 0909.3443Google Scholar
  10. 10.
    C. Li, On the limit behaviour of metrics in continuity method to Kähler–Einstein problem in toric Fano case. arxiv 1012.5229Google Scholar
  11. 11.
    R. Mazzeo, Kähler–Einstein metrics singular along a smooth divisor. J. Équ. aux Deriv. Partielles VI, 1–10 (1999)Google Scholar
  12. 12.
    R. Mazzeo, G. Montcouquiol, Infinitesimal rigidity of cone-manifolds and the Stoker problem for hyperbolic and Euclidean polyhdedra. arxiv 0908.2981Google Scholar
  13. 13.
    G. Montcouquiol, H. Weiss, Complex twist flows on surface group representations and the local structure of the deformation space of hyperbolic cone 3-manifolds. arxiv 1006.5582Google Scholar
  14. 14.
    J. Ross, R. Thomas, Weighted projective embeddings, stbility of orbifolds and constant scalar curvature Kähler metrics. J. Differ. Geom. 88, 109–159 (2011)Google Scholar
  15. 15.
    G. Szekelyhidi, Greatest lower bounds on the Ricci curvature of Fano manifolds. arxiv 0909.5504Google Scholar
  16. 16.
    G. Tian, S-T. Yau, Complete Kähler metrics with zero Ricci curvature I. J. Am. Math. Soc. 3, 579–609 (1990)Google Scholar
  17. 17.
    E.T. Whittaker, G.N. Watson, A Course in Modern Analysis (Cambridge University Press, Cambridge, 1927)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of MathematicsImperial CollegeLondonUK

Personalised recommendations