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October 2014 Structure of homogeneous Ricci solitons and the Alekseevskii conjecture
Ramiro Lafuente, Jorge Lauret
J. Differential Geom. 98(2): 315-347 (October 2014). DOI: 10.4310/jdg/1406552252

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.

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Ramiro Lafuente. Jorge Lauret. "Structure of homogeneous Ricci solitons and the Alekseevskii conjecture." J. Differential Geom. 98 (2) 315 - 347, October 2014. https://doi.org/10.4310/jdg/1406552252

Information

Published: October 2014
First available in Project Euclid: 28 July 2014

zbMATH: 1321.53051
MathSciNet: MR3263520
Digital Object Identifier: 10.4310/jdg/1406552252

Rights: Copyright © 2014 Lehigh University

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Vol.98 • No. 2 • October 2014
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