Classification of Nilsoliton metrics in dimension seven

Abstract

The aim of this paper is to classify Ricci soliton metrics on $7$-dimensional nilpotent Lie groups. It can be considered as a continuation of our paper in [Transformation Groups, Volume 17, Number 3 (2012), 639--656]. To this end, we use the classification of $7$-dimensional real nilpotent Lie algebras given by Ming-Peng Gong in his Ph.D thesis and some techniques from the results of Michael Jablonski in [Rocky Mtn. Journal of Math. 42 (2012), 1521--1549] and [Münster J. Math. 3 (2010), 67--88], and of Yuri Nikolayevsky in [Trans. Amer. Math. Soc. 363 (2011), 3935--3958]. Of the $9$ one-parameter families and $140$ isolated $7$-dimensional indecomposable real nilpotent Lie algebras, we have $99$ nilsoliton metrics given in an explicit form and $7$ one-parameter families admitting nilsoliton metrics. Our classification is the result of a case-by-case analysis, so many illustrative examples are carefully developed to explain how to obtain the main result.

21 pages. Some maple files with information on this paper are available in my homepage (sites.google.com/site/efernandezculma); anyone can use them (by crediting the author appropriately)

References in corpus (2)

• On distinguished orbits of reductive representations
• Laplacian flow of closed $G_2$-structures inducing nilsolitons

Cited by in corpus (7)

• Indefinite Einstein metrics on nice Lie groups
• Purely coclosed G${}_2$-structures on 2-step nilpotent Lie groups
• Indefinite nilsolitons and Einstein solvmanifolds
• On $G_2$-structures, special metrics and related flows
• Nice pseudo-Riemannian nilsolitons
• Ad-invariant metrics on nonnice nilpotent Lie algebras
• Codimension one Ricci soliton subgroups of solvable Iwasawa groups