[KSCI] Korea Science Citation Index Service

http://dx.doi.org/10.4134/JKMS.2007.44.1.109

LOCALLY HOMOGENEOUS CRITICAL METRICS ON FOUR-DIMENSIONAL MANIFOLDS

Kang, Yu-Tae (Department of Mathematics Sogang University)

Publication Information

Journal of the Korean Mathematical Society / v.44, no.1, 2007 , pp. 109-127 More about this Journal

Abstract

We classify complete, locally homogeneous metrics with finite volume on four-dimensional manifolds which are critical points for the squared $L^2-norm$ functionals of either the full Riemannian curvature tensor or the Weyl curvature tensor defined on the space of Riemannian metrics.

Keywords

locally homogeneous metric; critical metric; $L^2$ -norm curvature functional;

Citations & Related Records

Times Cited By Web Of Science : 1 (Related Records In Web of Science)
Times Cited By SCOPUS : 1

Reference
Cited By SCOPUS

1	F. Lamontagne, Critical metrics for the $L^2$ -norms of the curvature tensor, Ph. D. Thesis, 1993. Stony Brook
2	V. De Smedt and S. Salamon, Anti-self-dual metrics on Lie groups, Differential geometry and integrable systems (Tokyo, 2000), 63-75, Contemp. Math. 308, Amer. Math. Soc. Providence, RI, 2002
3	W. P. Thurston, Three-Dimensional Geometry and Topology Vol. 1., Edited by Silvio Levy, Princeton Mathematical Series, 35, Princeton Univ. Press, Princeton, New Jersey, 1997
4	A. Derdzinski, Self-dual Kahler manifolds and Einstein manifolds of dimension four, Compositio Math. 49 (1983), no. 3, 405-433
5	R. O. Filipkiewicz, Four dimensional geometries, Ph. D. Thesis, University of Warwick, 1984
6	F. Lamontagne, Une remarque sur la norme $L^2$ du tenseur de courbure, C. R. Acad. Sci. Paris Ser. I Math. 319 (1994), no. 3, 237-240
7	G. R. Jensen, Homogeneous Einstein spaces of dimension four, J. Differential Geometry 3 (1969), 309-349 DOI
8	J. Kim, Critical Kahler surfaces, Bull. Korean Math. Soc. 35 (1998), no. 3, 421-431
9	M. T. Anderson, Orbifold compactness for spaces of Riemannian metrics and applica- tions, Math. Ann. 331 (2005), no. 4, 739-778 DOI
10	A. L. Besse, Einstein manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 10, Springer-Verlag, 1987
11	C. LeBrun, Self-dual manifolds and hyperbolic geometry, Einstein metrics and Yang- Mills connections (Sanda, 1990), 99-131, Lecture Notes in Pure and Appl. Math., 145, Dekker, New York, 1993
12	J. Milnor, Curvatures of left invariant metrics on Lie groups, Advances in Math. 21 (1976), no. 3, 293-329 DOI
13	M. S. Raghunathan, Discrete subgroups of Lie groups, Springer-Verlag, N.Y., Heider- berg, Berlin, 1972
14	P. Scott, The geometries of 3-Manifolds, Bull. London Math. Soc. 15 (1983), no. 5, 401-487 DOI
15	G. Tian and J. Viaclovsky, Bach-flat asymptotically locally Euclidean metrics, Invent. Math. 160 (2005), no. 2, 357-415 DOI
16	P. Bernat, N. Conze, M. Duflo, M. Levy-Nahas, M. Rais, P. Renouard, and M. Vergne, Representations des groupes de Lie resolubles, Monographies de la Societe Mathematique de France, No. 4. Dunod, Paris, 1972
17	C. T. C. Wall, Geometric structures on compact complex analytic surfaces, Topology 25 (1986), no. 2, 119-153 DOI ScienceOn
18	C. T. C. Wall, Geometries and geometric structures in real dimension 4 and complex dimension 2, Lecture Notes in Math., 1167, Springer, Berlin, 1985, 268-292 DOI