Abstract. Following the approach of Bryant [10] we study the intrinsic torsion of a SU(3)-manifold deriving a number of formulae for the Ricci and the scalar curvature in terms of torsion forms. As a consequence we prove that in some special cases the Einstein condition forces the vanishing of the intrinsic torsion.

We characterize isometric actions on compact Kähler manifolds admitting a Lagrangian orbit, describing under which condition the Lagrangian orbit is unique. We furthermore give the complete classification of simple groups acting on the complex projective space with a Lagrangian orbit, and we give the explicit list of these orbits.

Abstract. We prove a general criterion to establish existence and uniqueness of a short-time solution to an evolution equation involving "closed" sections of a vector bundle, generalizing a method used by Bryant and Xu [8] for studying the Laplacian flow in G2-geometry. We apply this theorem in balanced geometry introducing a natural extension of the Calabi flow to the balanced case. We show that this flow has always a unique short-time solution belonging to the same Bott-Chern cohomology class of the initial balanced structure and that it preserves the Kähler condition. Finally we study explicitly the flow on the Iwasawa manifold.

Following the approach of Bryant [6], we study the intrinsic torsion of an SU(2)-structure on a 5-dimensional manifold deriving an explicit expression for the Ricci and the scalar curvature in terms of torsion forms and its derivative. As a consequence of this formula we prove that the α-Einstein condition forces some special SU(2)-structures to be Sasaki-Einstein.for r = 1, 2, 3, where the ω r 's are extended to exterior forms in the usual way.for any φ ∈ Λ 1 0 M and by J 1 (α) = J 2 (α) = J 3 (α) = 0 .Then for r = 1, 2, 3 one has • J 2 r = −I + R α ⊗ α ; • ω r (J r β, γ) = −ω r (β, J r γ) for every 1-form β, γ.Proof. The statement is a consequence of the real version of Schur's lemma (the proof is analogous to that of Proposition 2.1 of [2]).

We consider the natural generalization of the parabolic Monge-Ampère equation to HKT geometry. We prove that in the compact case the equation has always a short-time solution and when the hypercomplex manifold is locally flat and admits a hyperkähler metric, then the equation has a long-time solution whose normalization converges to a solution of the quaternionic Monge-Ampère equation first introduced in [4]. The result gives an alternative proof of a theorem of Alesker in [1].