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By Casey J.

ISBN-10: 1418185663

ISBN-13: 9781418185664

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5) Invariant Einstein metrics on certain Stiefel manifolds 41 If x13 = x23 = z then system (5) reduces to the following: x12 z 2 (k1 − k2 )x12 + (k2 − 1)x2 − (k1 − 1)x1 = 0, z2 ((k1 − 2)x2 − (k2 − 2)x1 )x212 + (k2 x1 − k1 x2 )x1 x2 z 2 + k3 (x1 − x2 )x1 x2 x212 = 0, z (k2 − 2)x212 + (k1 + k2 − 1)x22 − 2(k1 + k2 − 2)x2 x12 (6) + (k1 − 1)x1 x2 z 2 + k3 (x2 − x12 )x2 x212 = 0, z 2(k1 + k2 − 2)x12 − (k2 − 1)x2 − (k1 − 1)x1 z 2 + (k2 + k3 )x12 − 2(k3 + k2 + k1 − 2)z + (k1 − 1)x1 x212 = 0. From the first equation of system (6) we obtain that x1 = and substituting to the two other equations we obtain (k1 −k2 )x12 +(k2 −1)x2 , k1 −1 z(x2 − x12 ) ((2k2 + k1 − 2)x2 − (k2 − 2)x12 )z 2 + k3 x2 x212 = 0, z (k1 + 3k2 − 4)x12 − 2(k2 − 1)x2 z 2 + (k3 + k1 )x12 − 2(k1 + k2 + k3 − 2)z + (k2 − 1)x2 x212 = 0, (k1 − k2 )z 2 (x12 − x2 ) (1 − k1 )(k2 − 2)x212 + (k1 (k1 − 2) (7) + k2 (k1 − k2 ) + 1)x12 x2 + (k1 (k2 − 1) + k2 (k2 − 2) + 1)x22 z 2 + k3 (k1 − k2 )x12 + (k2 − 1)x2 x2 x212 = 0.

Albujer was partially supported by FPU Grant AP2004-4087 from Secretar´ıa de Estado de Universidades e Investigaci´on, MEC Spain and MEC/FEDER Grant MTM2004-04934-C04-02. References 1. L. Albujer, New examples of entire maximal graphs in H2 × R1 , to appear in Differential Geom. Appl. 2. L. J. 4363. Global behaviour of maximal surfaces in Lorentzian product spaces 33 3. L. J. Al´ıas, Parabolicity of maximal surfaces in Lorentzian product spaces, preprint 2007. 4. J. Al´ıas and B. Palmer, A duality result between the minimal surface equation and the maximal surface equation, An.

Ziller, Existence and Non-existence of Homogeneous Einstein Metrics, Invent. Math. 84 (1986) 177–194. 5. C. B¨ ohm, M. Wang and W. Ziller, A variational approach for compact homogeneous Einstein manifolds, Geom. Func. Anal. 14 (2004) 681–733. 6. C. B¨ ohm, Homogeneous Einstein metrics and simplicial complexes, J. Diff. Geom. 67 (2004) 79–165. 7. S. Kobayashi, Topology of positive pinched K¨ ahler manifolds, Tˆ ohoku Math. J. 15 (1963) 121–139. 8. A. Sagle, Some homogeneous Einstein manifolds, Nagoya Math.

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