Uniqueness of extremal Kähler metrics for an integral Kähler class. (English) Zbl 1058.32017
Let \(M\) be a compact, connected Kähler manifold and \([\omega]\) a fixed integral Kähler class in \(H^{1,1}(M)\). Let also \(H \subset Aut(M)\) be the maximal connected linear algebraic subgroup of the automorphism group of \(M\).
The author proves that if there exists an extremal metric \(\omega_o \in [\omega]\), then such a metric is unique up to a transformation in \(H\).
The proof is based on a recent result of the author in the paper [Osaka J. Math 41, No. 2, 463–472 (2004; Zbl 1070.58012)], where it was shown that any extremal metric in \([\omega]\) can be approximated in \({\mathcal C}^k\)-norm, for any given \(k\), by a suitable sequence of so-called “\(T\)-critical metrics”. The non existence of two distinct \(H\)-orbits of extremal metrics in \([\omega]\) is obtained from a non-trivial exploitation of a uniqueness property of the Chow points associated with \(T\)-critical metrics. The method of the proof is a clever development of arguments in [S. K. Donaldson, J. Differ. Geom. 59, 479–522 (2001; Zbl 1052.32017) and in X. X. Chen, J. Differ. Geom. 59, 189–234 (2000; Zbl 1041.58003)].
The author proves that if there exists an extremal metric \(\omega_o \in [\omega]\), then such a metric is unique up to a transformation in \(H\).
The proof is based on a recent result of the author in the paper [Osaka J. Math 41, No. 2, 463–472 (2004; Zbl 1070.58012)], where it was shown that any extremal metric in \([\omega]\) can be approximated in \({\mathcal C}^k\)-norm, for any given \(k\), by a suitable sequence of so-called “\(T\)-critical metrics”. The non existence of two distinct \(H\)-orbits of extremal metrics in \([\omega]\) is obtained from a non-trivial exploitation of a uniqueness property of the Chow points associated with \(T\)-critical metrics. The method of the proof is a clever development of arguments in [S. K. Donaldson, J. Differ. Geom. 59, 479–522 (2001; Zbl 1052.32017) and in X. X. Chen, J. Differ. Geom. 59, 189–234 (2000; Zbl 1041.58003)].
Reviewer: Andrea Spiro (Camerino)
MSC:
| 32Q20 | Kähler-Einstein manifolds |
| 14L24 | Geometric invariant theory |
| 53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |
Keywords:
Extremal Kähler metricsReferences:
| [1] | DOI: 10.1007/978-3-642-69828-6_8 · doi:10.1007/978-3-642-69828-6_8 |
| [2] | Calabi E., J. Differential Geom. 61 pp 173– · Zbl 1067.58010 · doi:10.4310/jdg/1090351383 |
| [3] | Chen X. X., J. Differential Geom. 56 pp 189– · Zbl 1041.58003 · doi:10.4310/jdg/1090347643 |
| [4] | Donaldson S. K., J. Differential Geom. 59 pp 479– · Zbl 1052.32017 · doi:10.4310/jdg/1090349449 |
| [5] | DOI: 10.1007/BF01446626 · Zbl 0831.53042 · doi:10.1007/BF01446626 |
| [6] | DOI: 10.1353/ajm.2000.0013 · Zbl 0972.53042 · doi:10.1353/ajm.2000.0013 |
| [7] | Mabuchi T., Osaka J. Math. 24 pp 227– |
| [8] | Mabuchi T., Osaka J. Math. 41 |
| [9] | D. Mumford, Questions on Algebraic Varieties, C.I.M.E., III Ciclo, Varenna, 1969 (Edizioni Cremonese, Rome, 1970) pp. 29–100. |
| [10] | DOI: 10.1007/978-3-642-57916-5_1 · doi:10.1007/978-3-642-57916-5_1 |
| [11] | DOI: 10.1007/s002220050176 · Zbl 0892.53027 · doi:10.1007/s002220050176 |
| [12] | Zelditch S., Int. Math. Res. Notices 6 pp 317– |
| [13] | Zhang S., Compositio Math. 104 pp 77– |
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