Convergence of homogeneous manifolds

Abstract

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.

26 pages, final version to appear in J. London Math. Soc

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• The Hull-Strominger system and the Anomaly flow on a class of solvmanifolds
• Positive Hermitian Curvature Flow on nilpotent and almost-abelian complex Lie groups