The Ricci flow in a class of solvmanifolds

ROMINA M. ARROYO

Córdoba, Argentina

Congreso; IV Congreso Latinoamericano de Matemáticos; 2012

The Ricci flow is a well-known evolution equation for a curve of Riemannian metrics on a manifold. In the case of Lie groups, it is equivalent in a natural and specific way to the bracket flow, that is an ordinary differential equation for a curve of Lie algebras. The objective of this communication is to analyze the Ricci flow of solvmanifolds whose Lie algebra has an abelian ideal of codimension 1; by using the bracket flow. We have proved that the forward time interval for the flow is [0;infinity); the omega-limit is a single point, i.e. there is no ?chaos?, and that the Ricci flow converges in the pointed sense to a manifold which is locally isometric to a flat manifold. To prevent some solutions from converging to a flat metric, we have studied a normalized bracket flow. We give a monotone decreasing function that will allow us to prove that limits of subsequences are algebraic solitons, and to determine which of these solutions converge to flat metrics. Finally, we will use these results to prove that if a Lie group in this class admits a Riemannian metric of negative sectional curvature, then the curvature of any Ricci flow solution will become negative before the first singularity time.