Finding solitons

Abstract

On each solvable Lie group, there is at most one solv- soliton up to isometry and scaling. This allows us to en- dow several Lie groups that do not admit Einstein metrics (e.g., nilpotent or unimodular solvable Lie groups) with a canonical Riemannian metric. Analogously, Chern?Ricci, pluriclosed, and HCF (resp., SCF) algebraic solitons pro- vide distinguished Hermitian (resp., almost-Kähler) struc- tures for Lie groups on which Kähler metrics do not exist. Laplacian algebraic solitons play the same role in the ho- mogeneous case, where holonomy 2 is out of reach since Ricci at implies at.The moving-bracket approach allows the rich interplay between soliton geometric structures on Lie groups and soliton Lie algebras, paving the way to many beautiful ap- plications of GIT to differential geometry.