Structure of homogeneous Ricci solitons and the Alekseevskii conjecture

• Autores: Lafuente, Ramiro Augusto; Lauret, Jorge Ruben
• Año de publicación: 2014
• Idioma: inglés
• Tipo de recurso: artículo
• Materia: RICCI SOLITONS; HOMOGENEOUS MANIFOLDS; ALEKSEEVSKII CONJECTURE
• Institución: Consejo Nacional de Investigaciones Científicas y Técnicas

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/KM=G/K is diffeomorphic to a product U/K×NU/K×N, where UU is a maximal reductive Lie subgroup of GG and NN is the maximal nilpotent normal subgroup of GG, such that the metric restricted to NN is a nilsoliton. Moreover, strong compatibility conditions between the metric and the action of UU on NN by conjugation must hold, including a nice formula for the Ricci operator of the metric restricted to U/KU/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.
Fil: Lafuente, Ramiro Augusto. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Córdoba. Centro de Investigación y Estudios de Matemática de Córdoba(p); Argentina. Universidad Nacional de Córdoba; Argentina
Fil: Lauret, Jorge Ruben. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Córdoba. Centro de Investigación y Estudios de Matemática de Córdoba(p); Argentina. Universidad Nacional de Córdoba; Argentina