On homogeneous Ricci solitons

RAMIRO LAFUENTE; JORGE LAURET

QUARTERLY JOURNAL OF MATHEMATICS

OXFORD UNIV PRESS

Lugar: Oxford; Año: 2014 vol. 65 p. 399 - 419

0033-5606

We study the evolution of homogeneous Ricci solitons under the bracket flow, a dynamical system on the space $hca_{q,n}subsetlamg$ 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 $omega$-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.