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.

Symplectic forms taming complex structures on compact manifolds are strictly related to Hermitian metrics having the fundamental form ∂∂-closed, i.e., to strong Kähler with torsion (SKT) metrics. It is still an open problem to exhibit a compact example of a complex manifold having a tamed symplectic structure but non-admitting Kähler structures. We show some negative results for the existence of symplectic forms taming complex structures on compact quotients of Lie groups by discrete subgroups. In particular, we prove that if M is a nilmanifold (not a torus) endowed with an invariant complex structure J, then (M, J) does not admit any symplectic form taming J. Moreover, we show that if a nilmanifold M endowed with an invariant complex structure J admits an SKT metric, then M is at most 2-step. As a consequence we classify eight-dimensional nilmanifolds endowed with an invariant complex structure admitting an SKT metric.

We obtain some general results on Sasakian Lie algebras and prove as a consequence that a (2n + 1)-dimensional nilpotent Lie group admitting left-invariant Sasakian structures is isomorphic to the real Heisenberg group H 2n+1 . Furthermore, we classify Sasakian Lie algebras of dimension five and determine which of them carries a Sasakian α-Einstein structure. We show that a five-dimensional solvable Lie group with a left-invariant Sasakian structure and which admits a compact quotient by a discrete subgroup is isomorphic to either H 5 or a semidirect product R (H 3 × R). In particular, the compact quotient is an S 1 -bundle over a four-dimensional Kähler solvmanifold.

We prove the following quantitative version of the celebrated Soap Bubble Theorem of Alexandrov. Let S be a C 2 closed embedded hypersurface of R n+1 , n ≥ 1, and denote by osc(H) the oscillation of its mean curvature. We prove that there exists a positive ε, depending on n and upper bounds on the area and the C 2 -regularity of S, such that if osc(H) ≤ ε then there exist two concentric balls Br i and Br e such that S ⊂ Br e \ Br i and re − ri ≤ C osc(H), with C depending only on n and upper bounds on the surface area of S and the C 2 regularity of S. Our approach is based on a quantitative study of the method of moving planes and the quantitative estimate on re − ri we obtain is optimal.As a consequence of this theorem, we also prove that if osc(H) is small then S is diffeomorphic to a sphere and give a quantitative bound which implies that S is C 1 -close to a sphere.

We study evolution of (strong Kähler with torsion) SKT structures via the pluriclosed flow on complex nilmanifolds, i.e. on compact quotients of simply connected nilpotent Lie groups by discrete subgroups endowed with an invariant complex structure. Adapting to our case the techniques introduced by Jorge Lauret for studying Ricci flow on homogeneous spaces we show that for SKT Lie algebras the pluriclosed flow is equivalent to a bracket flow and we prove a long time existence result in the nilpotent case. Finally, we introduce a natural flow for evolving tamed symplectic forms on a complex manifold, by considering evolution of symplectic forms via the flow induced by the Bismut Ricci form.