Structure of homogeneous Ricci solitons and the Alekseevskii conjecture
By Ramiro A. Lafuente, Jorge Lauret. 2012
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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 x 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.
Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/b3bef7a19bd3ed8e5e62e42bd795047bdd3ee641
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In Applied Mathematics
In Geometry and Topology
In Mathematical Physics
Citations in the corpus
- Homogeneous Ricci Solitons are Algebraic (2013, methodology): “The theorem above resolves questions raised by Lafuente-Lauret [12] and He-Petersen-Wylie [4].”