By Lauret J.

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Reine angew. Math. 530 (2001), 17–31. 25. : On the real moment map, Math. Res. Lett. 8 (2001), 779–788. 26. , Fogarty, J. : Geometric Invariant Theory, 3rd edn, Berlin-Heidelberg: Springer Verlag (1994). 27. : A stratification of the null cone via the momentum map, Amer. J. Math. 106 (1984), 1281–1329 (with an appendix by D. Mumford). 28. Richardson, R. W. and Slodowy, P. : Minimum vectors for real reductive algebraic groups, J. London Math. Soc. 42(2) (1990), 409–429. 29. : Complex structures on nilpotent Lie algebras, J.

X n } is any orthonormal basis of (n, ·, · ). Notice that always sc(N , ·, · ) < 0, unless N is Abelian. It is proved in [16, Lemma 1] (see also [15]) that the gradient of the scalar curvature functional sc: P → R (recall that the scalar curvature is left invariant and so it consists in a single constant for each metric) is given in the nilpotent case by grad(sc) ·,· = − ric ·,· , (29) and hence it follows from the properties of P described above that tr Ric ·,· D = 0, ∀ symmetric D ∈ Der(n), (30) where Der(n) is the Lie algebra of derivations of n (see for instance [20, (2)] for a proof of this fact).

184 (2003), 41–57. 13. : Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982), 255–306. 14. : The formation of singularities in the Ricci flow, Surv. in Differential Geom. II (1995), 7–136. 15. : Noncompact homogeneous Einstein spaces, Invent. Math. 133 (1998), 279–352. 16. : The scalar curvature of left-invariant Riemannian metrics, Indiana Math. J. 20 (1971), 1125–1144. 17. : Cohomology of quotients in symplectic and algebraic geometry, Mathematical Notes 31 (1984), Princeton University Press, Princeton.

### A Canonical Compatible Metric for Geometric Structures on Nilmanifolds by Lauret J.

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