|
Hermitian-Einstein metrics for vector bundles on complete Kähler manifolds
Author(s):
Lei
Ni;
Huaiyu
Ren
Journal:
Trans. Amer. Math. Soc.
353
(2001),
441-456.
MSC (2000):
Primary 58G11
Posted:
August 3, 2000
Retrieve article in:
PDF
This article is available free of charge
Abstract |
References |
Similar articles |
Additional information
Abstract:
In this paper, we prove the existence of Hermitian-Einstein metrics for holomorphic vector bundles on a class of complete Kähler manifolds which include Hermitian symmetric spaces of noncompact type without Euclidean factor, strictly pseudoconvex domains with Bergman metrics and the universal cover of Gromov hyperbolic manifolds etc. We also solve the Dirichlet problem at infinity for the Hermitian-Einstein equations on holomorphic vector bundles over strictly pseudoconvex domains.
References:
-
- [A-S]
- M. Anderson & R. Schoen, Positive harmonic functions on complete manifolds of negative curvature, Annals of Math. 121(1985), 429-461. MR 87a:58151
- [A-C-M]
- P. Avilés, H. I. Choi and M. Micallef, Boundary behavior of harmonic maps on nonsmooth domains and complete negatively curved manifolds, J. Funct. Anal. 99(1991), 293-331. MR 92j:58025
- [B]
- S. Bando, Einstein-Hermitian metrics on noncompact Kähler manifolds, Einstein metrics and Yang-Mills connections (Sanda, 1990), 27-33, Lecture Notes in Pure and Appl. Math. Dekker, New York, 1993. MR 94d:32040
- [B-S]
- S. Bando & S. T. Siu, Stable sheaves and Einstein-Hermitian metrics, Geometry and analysis on complex manifolds, 39-50, World. Sci. Publishing, River Edge, NY, 1994. MR 98e:32053
- [C-Y]
- S. Y. Cheng & S. T. Yau, On the existence of a complete Kähler manifolds and the regularity of Fefferman's equation, Commun. Pure Appl. Math. 33(1980), 507-544. MR 82f:53074
- [D1]
- S. K. Donaldson, Anti Self-Dual Yang-Mills Connections over Complex Algebraic Surfaces and Stable Vector Bundle, Proc. London Math. Soc. (3), 50(1985), 1-26. MR 86h:58038
- [D2]
- S. K. Donaldson, Boundary value problems for Yang-Mills fields, Journal of Geometry and Physics, 8(1992), 89-122. MR 93d:53033
- [D3]
- S. K. Donaldson, Infinite determinants, stable bundles and curvature, Duke Math. J.54(1987), 231-147. MR 88g:32046
- [Do]
- H. Donnelly,
 cohomology of pseudoconvex domains with complete Kähler metrics, Michigan Math. J. 41(1994), 433-442. MR 95h:32007 - [G-H]
- P. Griffiths & J. Harris, Principles of Algebraic Geometry, John Wiley, New York, 1978. MR 80b:14001
- [G-T]
- D. Gilbarg & N. S. Trudinger, Elliptic partial differential equations of second order, 2nd edition, Springer-Verlag, 1983. MR 86c:35035
- [Gri]
- A. Grigor'yan, Heat kernel upper bounds on a complete noncompact manifold, Rev. Math. Iberoamericana. 10(1994), 395-452 MR 96b:58017
- [Gro]
- M. Gromov, Kähler hyperbolicity and
 -Hodge theory, J. Diff. Geom. 33(1991), 263-292. MR 92a:58133 - [Guo]
- G-Y Guo, Yang-Mills fields on cylindrical manifolds and holomorphic bundles I, Commun. Math. Phys. 179(1996), 737-775. MR 98a:32040
- [K]
- S. Kobayashi, Differential Geometry of Complex Vector Bundles, Princeton, 1986. MR 89e:53100
- [L1]
- P. Li, Lecture notes on geometric analysis, Research Institute of Mathematics, Global Analysis Center Seoul National University, 1993. MR 96m:58269
- [L-T]
- P. Li & L. F. Tam, The heat equation and harmonic maps of complete manifolds, Invent. Math. 105(1991), 1-46. MR 93e:58039
- [N1]
- L. Ni, Vanishing theorems on complete Kähler manifolds and their applications, J. Diff. Geom. 50(1998), 89-122. CMP 99:10
- [N2]
- L. Ni, Hermitian harmonic maps from complete Hermitian manifolds to complete Riemannian manifolds, Math. Z. 232(1999), 331-355. CMP 2000:04
- [N-S]
- M. S. Narasimhan & C. S. Seshadri, Stable and unitray vector bundles on a compact Riemann surface, Annals of Math. 82(1965), 540-564. MR 32:1725
- [Si]
- C. T. Simpson, Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization, JAMS. 1 (1988), 867-918. MR 90e:58026
- [Siu]
- Y. T. Siu, Lectures on Hermitian-Einstein metrics for stable bundles and Kähler-Einstein metrics , Birkhäuser, Basel-Boston, 1987. MR 89d:32020
- [S-Y]
- R. Schoen & S. T. Yau, Lectures on Differential Geometry, Academia Sinica Press, Beijing, 1988; English transl., Internat. Press, Cambridge, MA, 1994. MR 97d:53001
- [U-Y]
- K. Uhlenbeck & S. T. Yau, On the existence of Hermitian-Yang-Mills connections in stable vector bundles, Comm. Pure Appl. Math. 39(1986), 257-293. MR 88c:58154
Similar Articles:
Retrieve articles in Transactions of the American Mathematical Society
with MSC
(2000):
58G11
Retrieve articles in all Journals with MSC
(2000):
58G11
Additional Information:
Lei
Ni
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Address at time of publication:
Department of Mathematics, Stanford University, Stanford, California 94305
Email:
lni@math.purdue.edu
Huaiyu
Ren
Affiliation:
Department of Mathematics, University of California, Irvine, California 92697
Email:
hren@math.uci.edu
DOI:
10.1090/S0002-9947-00-02549-6
PII:
S 0002-9947(00)02549-6
Received by editor(s):
November 5, 1998
Received by editor(s) in revised form:
March 5, 1999
Posted:
August 3, 2000
Additional Notes:
Research was partially supported by an NSF grant
Copyright of article:
Copyright
2000,
American Mathematical Society
|
Comments: webmaster@ams.org