Structure of homogeneous Ricci solitons and the Alekseevskii conjecture

Abstract

We bring new insights into the longstanding 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 $U / K \times N$, 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.