Curvature flows for almost-hermitian Lie groups

We study curvature flows in the locally homogeneous case (e.g. compact quotients of Lie groups, solvmanifolds, nilmanifolds) in a unified way, by considering a generic flow under just a few natural conditions on the broad class of almost-hermitian structures. As a main tool, we use an ODE system defined on the variety of 2n-dimensional Lie algebras, called the bracket flow, whose solutions differ from those to the original curvature flow by only pull-back by time-dependent diffeomorphisms. The approach, which has already been used to study the Ricci flow on homogeneous manifolds, is useful to better visualize the possible pointed limits of solutions, under diverse rescalings, as well as to address regularity issues. Immortal, ancient and self-similar solutions arise naturally from the qualitative analysis of the bracket flow. The Chern-Ricci flow and the symplectic curvature flow are considered in more detail.Comment: 25 pages. Final version to appear in Transactions AM

Examples of Anosov diffeomorphisms

We give a simple procedure to construct explicit examples of nilmanifolds admitting an Anosov diffeomorphism, and show that a reasonable classification up to homeomorphism (or even up to commensurability) of such nilmanifolds would not be possible.Comment: 8 pages, AMS-LaTE

Convergence of homogeneous manifolds

We study in this paper three natural notions of convergence of homogeneous manifolds, namely infinitesimal, local and pointed, and their relationship with a fourth one, which only takes into account the underlying algebraic structure of the homogeneous manifold and is indeed much more tractable. Along the way, we introduce a subset of the variety of Lie algebras which parameterizes the space of all n-dimensional simply connected homogeneous spaces with q-dimensional isotropy, providing a framework which is very advantageous to approach variational problems for curvature functionals as well as geometric evolution equations on homogeneous manifolds.Comment: 26 pages, final version to appear in J. London Math. So