The space of Solvsolitons in low dimensions

WILL, CYNTHIA

Cordoba

Congreso; IV Congreso Latinoamericano de -matemáticos; 2012

FaMAF-UMALCA

The simultaneous diagonalization of the Ricci tensor has been crucial in the study of the Ricci ?ow on the 3-dimensional homogeneous manifolds (see [2, 1, 3, 4, 6]) from the Geometrization Conjecture. The lack of these special bases in dimension ≥ 4 has obstructed the study of the Ricci Flow for several left-invariant metrics. It is therefore natural to ask which Lie algebras admit a stably Ricci-diagonal basis, in the sense that any diagonal left-invariant metric has diagonal Ricci tensor, or even farther, which left-invariant metrics admit an orthonormal stably Ricci-diagonal basis. In this talk, we shall prove that a basis of a nilpotent Lie algebra is stably Ricci-diagonal if and only if it produces quite a simple set of structural constants (so called nice bases in the literature). We also present solvable and semisimple examples to show that the nilpotent condition is necessary in both directions of the result. As an application, we obtain that already in dimension 4, there are nilmanifolds whose Ricci Flow is not diagonal with respect to any orthonormal basis. We also prove that any left-invariant (algebraic) Ricci soliton on a non-necessarily nilpotent Lie group, gives rise to a diagonal Ricci Flow solution.