Hyperkähler metrics of cohomogeneity one. (English) Zbl 0909.53032
The intractability of the Einstein equations has led mathematicians interested in Einstein metrics to impose simplifying assumptions. One such assumption is that of homogeneity, when the Einstein condition may be expressed purely algebraically. Another natural restriction is to consider metrics of cohomogeneity one, that is, where the generic orbit of the isometry group has real codimension one. For cohomogeneity-one metrics the Einstein condition becomes a system of non-linear ordinary differential equations. Although considerable progress has been made, a classification of complete cohomogeneity-one Einstein metrics still seems far off.
The authors prove a classification theorem for cohomogeneity-one metrics which satisfy the stronger condition of being hyperkähler: Let \(M\) be an irreducible hyperkähler manifold of dimension greater than 4 and of cohomogeneity one with respect to a compact simple Lie group \(G\). Then \(M\) is an open subset of the cotangent bundle of complex projective space with the Calabi metric [E. Calabi, Ann. Sci. Éc. Norm. Supér., IV. Sér. 12, 269-294 (1979; Zbl 0431.53056)], or the Swann space \(\mathcal U(N)\) [A. F. Swann, Math. Ann. 289, 421-450 (1991; 718.53051)] (with its standard metric), where \(N\) is a compact Wolf space. If \(M\) is complete, then \(M\) is \(T^\ast\mathbb C\)P\((n)\) with the Calabi metric.
The authors prove a classification theorem for cohomogeneity-one metrics which satisfy the stronger condition of being hyperkähler: Let \(M\) be an irreducible hyperkähler manifold of dimension greater than 4 and of cohomogeneity one with respect to a compact simple Lie group \(G\). Then \(M\) is an open subset of the cotangent bundle of complex projective space with the Calabi metric [E. Calabi, Ann. Sci. Éc. Norm. Supér., IV. Sér. 12, 269-294 (1979; Zbl 0431.53056)], or the Swann space \(\mathcal U(N)\) [A. F. Swann, Math. Ann. 289, 421-450 (1991; 718.53051)] (with its standard metric), where \(N\) is a compact Wolf space. If \(M\) is complete, then \(M\) is \(T^\ast\mathbb C\)P\((n)\) with the Calabi metric.
Reviewer: Serguey M.Pokas (Odessa)
MSC:
| 53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |
| 57S25 | Groups acting on specific manifolds |
| 22E60 | Lie algebras of Lie groups |
References:
| [1] | Alekseevskiǐ, D. V., Functional Anal. Appl., 2, 106-114 (1968), English translation: · Zbl 0238.53040 |
| [2] | Alekseevskiǐ, D. V., Functional. Anal. Appl., 4, 321-322 (1970), English translation: · Zbl 0238.53040 |
| [3] | Atiyah, M. F.; Hitchin, N. J., The Geometry and Dynamics of Magnetic Monopoles (1988), Princeton University Press: Princeton University Press Princeton · Zbl 0671.53001 |
| [4] | Belinskii, V.; Gibbons, G. W.; Page, D. N.; Pope, C. N., Asymptotically Euclidean metrics in quantum gravity, Phys. Lett. B, 76, 433-435 (1978) |
| [5] | Bérard-Bergery, L., Sur de nouvelles variétés riemanniennes d’Einstein (1982), Publications de l’Institut E. Cartan: Publications de l’Institut E. Cartan Nancy · Zbl 0544.53038 |
| [6] | Besse, A., Einstein Manifolds (1988), Springer: Springer Berlin |
| [7] | Bielawski, R., Invariant hyperkähler metrics with a homogeneous complex structure, McMaster University preprint (1996) |
| [8] | Biquard, O., Sur les équations de Nahm et la structure de Poisson des algèbres de Lie semi-simples complexes (1994), preprint |
| [9] | Bogoyavlenskii, O. I., Methods in the Qualitative Theory of Dynamical systems in Astrophysics and Gas Dynamics, (Springer Ser. Soviet Math. (1985), Springer: Springer Berlin) · Zbl 0774.35013 |
| [10] | Calabi, E., Metriques Kahleriennes et fibrès holomorphes, Ann. Sci. Ecole Norm. Sup., 12, 269-294 (1979) · Zbl 0431.53056 |
| [11] | Cheeger, J.; Gromoll, D., The splitting theorem for manifolds of non-negative Ricci curvature, J. Differential Geom., 6, 119-128 (1971) · Zbl 0223.53033 |
| [12] | Dancer, A. S.; Strachan, I. A.B., Kähler-Einstein metrics with SU (2) action, (Math. Proc. Cambridge Philos. Soc., 115 (1994)), 513-525 · Zbl 0811.53046 |
| [13] | J. Eschenburg and M. Wang, The initial value problem for cohomogeneity one Einstein metrics (preprint).; J. Eschenburg and M. Wang, The initial value problem for cohomogeneity one Einstein metrics (preprint). · Zbl 0992.53033 |
| [14] | Gibbons, G. W.; Pope, C. N., The positive action conjecture and asymptotically Euclidean metrics in quantum gravity, Comm. Math. Phys., 66, 267-290 (1979) |
| [15] | Humphrey, J. E., Introduction to Lie algebras and Representation Theory (1972), Springer: Springer Berlin · Zbl 0254.17004 |
| [16] | Kovalev, A. G., Nahm’s equations and complex adjoint orbits, Quart. J. Math. Oxford Ser., 2, 47, 41-58 (1996) · Zbl 0852.53033 |
| [17] | Lichnerowicz, A., Variétés de Jacobi et espaces homogènes de contact complexes, J. Math. Pures. Appl., 67, 131-173 (1988) · Zbl 0644.53032 |
| [18] | Swann, A. F., HyperKähler and quaternionic Kähler geometry, Math. Ann., 289, 421-450 (1991) · Zbl 0711.53051 |
| [19] | Wolf, J. A., Complex homogeneous contact manifolds and quaternionic symmetric spaces, J. Math. Mech., 14, 1033-1047 (1965) · Zbl 0141.38202 |
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