Structure of homogeneous Ricci solitons and the Alekseevskii conjecture

RAMIRO A. LAFUENTE; JORGE LAURET

JOURNAL OF DIFFERENTIAL GEOMETRY

INT PRESS BOSTON, INC

Año: 2014 vol. 98 p. 315 - 347

0022-040X

We bring new insights into the long-standing Alekseevskii conjecture, namely that any connected homogeneous Einstein manifold of negative scalar curvature is diffeomorphic to a Euclidean space, by proving structural results which are actually valid for any homogeneous expanding Ricci soliton, and generalize many well-known results on Einstein solvmanifolds, solvsolitons and nilsolitons. We obtain that any homogeneous expanding Ricci soliton $M=G/K$ is diffeomorphic to a product $U/K imes N$, where $U$ is a maximal reductive Lie subgroup of $G$ and $N$ is the maximal nilpotent normal subgroup of $G$, such that the metric restricted to $N$ is a nilsoliton. Moreover, strong compatibility conditions between the metric and the action of $U$ on $N$ by conjugation must hold, including a nice formula for the Ricci operator of the metric restricted to $U/K$. Our main tools come from geometric invariant theory. As an application, we give many Lie theoretical characterizations of algebraic solitons, as well as a proof of the fact that the following a priori much stronger result is actually equivalent to Alekseevskii´s conjecture: Any expanding algebraic soliton is diffeomorphic to a Euclidean space.