Kähler-Einstein metrics singular along a smooth divisor
Journées équations aux dérivées partielles (1999), article no. 6, 10 p.

In this note we discuss some recent and ongoing joint work with Thalia Jeffres concerning the existence of Kähler-Einstein metrics on compact Kähler manifolds which have a prescribed incomplete singularity along a smooth divisor D. We shall begin with a general discussion of the problem, and give a rough outline of the “classical” proof of existence in the smooth case, due to Yau and Aubin, where no singularities are prescribed. Following this is a discussion of the geometry of the conical or edge singularities and then some discussion of the new elements of the proof in this context.

@incollection{JEDP_1999____A6_0,
     author = {Raffe Mazzeo},
     title = {K\"ahler-Einstein metrics singular along a smooth divisor},
     booktitle = {},
     series = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {6},
     pages = {1--10},
     year = {1999},
     publisher = {Universit\'e de Nantes},
     doi = {10.5802/jedp.550},
     zbl = {01810579},
     language = {en},
     url = {https://proceedings.centre-mersenne.org/articles/10.5802/jedp.550/}
}
TY  - JOUR
AU  - Raffe Mazzeo
TI  - Kähler-Einstein metrics singular along a smooth divisor
JO  - Journées équations aux dérivées partielles
PY  - 1999
SP  - 1
EP  - 10
PB  - Université de Nantes
UR  - https://proceedings.centre-mersenne.org/articles/10.5802/jedp.550/
DO  - 10.5802/jedp.550
LA  - en
ID  - JEDP_1999____A6_0
ER  - 
%0 Journal Article
%A Raffe Mazzeo
%T Kähler-Einstein metrics singular along a smooth divisor
%J Journées équations aux dérivées partielles
%D 1999
%P 1-10
%I Université de Nantes
%U https://proceedings.centre-mersenne.org/articles/10.5802/jedp.550/
%R 10.5802/jedp.550
%G en
%F JEDP_1999____A6_0
Raffe Mazzeo. Kähler-Einstein metrics singular along a smooth divisor. Journées équations aux dérivées partielles (1999), article  no. 6, 10 p.. doi: 10.5802/jedp.550

[1] J.-P. Bourguignon, éd Preuve de la Conjecture de Calabi, Astérisque Vol. 58, Soc. Math. France, Paris (1978).

[2] A. Byde, Thesis, Stanford University (2000). In preparation.

[3] S.Y. Cheng and S.T. Yau, On the existence of complete Kähler metrics on non-compact complex manifolds and the regularity of Fefferman's equation, Comm. Pure Appl. Math. 33 No. 4 (1980), 507-544. | MR | Zbl

[4] C. Fefferman, The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math. 26 (1974), 1-65. | MR | Zbl

[5] C. Hodgson and S. Kerckhoff, Rigidity of hyperbolic cone manifolds and hyperbolic Dehn surgery, Jour. Diff. Geom. 48 (1998), 1-59. | MR | Zbl

[6] D. Joyce, Asymptotically locally Euclidean metrics with holonomy SU(m), XXX Mathematics Archive Preprint AG/9905041. | Zbl

[7] D. Joyce, Quasi-ALE metrics with holonomy SU(m) and Sp(m), XXX Mathematics Archive Preprint AG/9905043. | Zbl

[8] R. Kobayashi, Einstein-Kähler V metrics on open Satake V -surfaces with isolated quotient singularities, Math. Ann. 272 (1985), 385-398. | MR | Zbl

[9] J. Lee and R. Melrose, Boundary behaviour of the complex Monge-Ampere equation, Acta Math. 148 (1982), 159-192. | MR | Zbl

[10] R. Mcowen, Point singularities and conformal metrics on Riemann surfaces, Proc. Amer. Math. Soc. 103 No. 1 (1988), 222-224. | MR | Zbl

[11] R. Mazzeo, Elliptic theory of differential edge operators I, Comm. Partial Diff. Eq., 16 No. 10 (1991), 1616-1664. | MR | Zbl

[12] R. Mazzeo, Regularity for the singular Yamabe problem, Indiana Univ. Math. Jour. 40 (1991), 1277-1299. | MR | Zbl

[13] B.-W. Schulze, Boundary value problems and singular pseudo-differential operators. Wiley, Chichester - New York, 1998. | MR | Zbl

[14] G. Tian and S.T. Yau, Existence of Kähler-Einstein metrics on complete Kähler manifolds and their applications to algebraic geometry, in Mathematical Aspects of String Theory (S.T. Yau, ed.), World Scientific (1987), 574-628. | MR | Zbl

[15] G. Tian and S.T. Yau, Complete Kähler manifolds with zero Ricci curvature I, J. Amer. Math. Soc., 3 No. 3 (1990), 579-609. | MR | Zbl

[16] G. Tian and S.T. Yau, Complete Kähler manifolds with zero Ricci curvature II, Invent. Math. 106 No. 1 (1991), 27-60. | MR | Zbl

[17] M. Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 324 No. 2 (1991), 793-821. | MR | Zbl

Cité par Sources :