We bring new insights into the long-standing Alekseevskii conjecture, namely that any connected homogeneous Einstein manifold of negative scalar curvature is diffeomorphic to a Euclidean space, by proving structural results which are actually valid for any homogeneous expanding Ricci soliton, and generalize many well-known results on Einstein solvmanifolds, solvsolitons and nilsolitons. We obtain that any homogeneous expanding Ricci soliton M = G/K is diffeomorphic to a product U/K × N , where U is a maximal reductive Lie subgroup of G and N is the maximal nilpotent normal subgroup of G, such that the metric restricted to N is a nilsoliton. Moreover, strong compatibility conditions between the metric and the action of U on N by conjugation must hold, including a nice formula for the Ricci operator of the metric restricted to U/K. Our main tools come from geometric invariant theory. As an application, we give many Lie theoretical characterizations of algebraic solitons, as well as a proof of the fact that the following a priori much stronger result is actually equivalent to Alekseevskii's conjecture: Any expanding algebraic soliton is diffeomorphic to a Euclidean space.

We study the asymptotic behavior of the pluriclosed flow in the case of left‐invariant Hermitian structures on Lie groups. We prove that solutions on 2‐step nilpotent Lie groups and on almost‐abelian Lie groups converge, after a suitable normalization, to self‐similar solutions of the flow. Given that the spaces are solvmanifolds, an unexpected feature is that some of the limits are shrinking solitons. We also exhibit the first example of a homogeneous manifold on which a geometric flow has some solutions with finite extinction time and some that exist for all positive times.

Abstract. We prove uniform curvature estimates for homogeneous Ricci flows: For a solution defined on [0, t] the norm of the curvature tensor at time t is bounded by the maximum of C(n)/t and C(n)(scal(g(t)) − scal(g(0))). This is used to show that solutions with finite extinction time are Type I, immortal solutions are Type III and ancient solutions are Type I, where all the constants involved depend only on the dimension n. A further consequence is that a non-collapsed homogeneous ancient solution on a compact homogeneous space emerges from a unique Einstein metric on the same space.The above curvature estimates are proved using a gap theorem for Ricciflatness on homogeneous spaces. The proof of this gap theorem is by contradiction and uses a local W 2,p convergence result, which holds without symmetry assumptions.The proof of Thurston's geometrization conjecture by Perelman [Per1], [Per2], [Per3] using Hamilton's Ricci flow [Ha] can certainly be considered a major break through. There are however interesting related problems which remain open. For instance, Lott asked in [Lo2], whether the 3-dimensional Ricci flow detects the homogeneous pieces in the geometric decomposition proposed by Thurston. In the same paper this was proved to be true for immortal solutions, assuming a Type III behavior of the curvature tensor and a natural bound on the diameter of the underlying closed oriented manifold. More recently, Bamler showed in a series of papers that for the Ricci flow with surgery there exist only finitely many surgery times, and that the Type III behavior holds after the last surgery time. In many cases, convergence to a geometric piece could be established. We refer to [Bam] and the papers quoted therein.Recall that a Ricci flow solution is called homogeneous, if it is homogeneous at every time. In dimension 3 homogeneous Ricci flows are well understood: see [IJ]

We show that for an immortal homogeneous Ricci flow solution any sequence of parabolic blow-downs subconverges to a homogeneous expanding Ricci soliton. This is established by constructing a new Lyapunov function based on curvature estimates which come from real geometric invariant theory.In 1982 Hamilton showed that on a compact 3-dimensional manifold any metric of positive Ricci curvature can be deformed to the round metric via volume normalized Ricci flow [Ham82]. Similar convergence results were obtained afterwards in higher dimensions, see for instance [Ham86], [BW08] and [BS09]. About 20 years later, Perelman was able to describe new geometric quantities, which are monotone along Ricci flow solutions [Per02], [Per03]. In the case of finite extinction time, assuming a Type-I behavior of the evolved curvature tensors, it could then be shown that any essential blow-up sequence subconverges to a nonflat, gradient Ricci soliton [Nab10], [EMT11].By contrast, for immortal Ricci flow solutions, assuming a Type-III behavior of the evolved curvature tensors, subconvergence of an essential blow-down sequence to an expanding limit soliton is unknown in general. The low-dimensional situation is however much better understood: in dimension two the existence of a locally homogeneous limit solution of constant curvature could be established [Ham88], [Cho91], [JMS09], and in dimension three subconvergence to an expanding locally homogeneous limit soliton could be shown in several cases [Lot10], [Bam14].

Abstract. We study the evolution of homogeneous Ricci solitons under the bracket flow, a dynamical system on the space Hq,n ⊂ Λ 2 g * ⊗ g of all homogeneous spaces of dimension n with a q-dimensional isotropy, which is equivalent to the Ricci flow for homogeneous manifolds. We prove that algebraic solitons (i.e. the Ricci operator is a multiple of the identity plus a derivation) are precisely the fixed points of the system, and that a homogeneous Ricci soliton is isometric to an algebraic soliton if and only if the corresponding bracket flow solution is not chaotic, in the sense that its ω-limit set consists of a single point. We also geometrically characterize algebraic solitons among homogeneous Ricci solitons as those for which the Ricci flow solution is simultaneously diagonalizable.