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 [M\"{u}nster 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.