Explicit Ricci solitons on nilpotent Lie groups

TL;DR

This paper explicitly constructs Ricci solitons on higher-dimensional Heisenberg and unitriangular matrix groups, demonstrating convergence of metrics and providing concrete examples of solitons on these nilpotent Lie groups.

Contribution

It provides explicit constructions of Ricci solitons on classes of nilpotent Lie groups, extending known results and employing novel methods like Lott's blowdown and specific ansatz.

Findings

Explicit Ricci solitons constructed on higher-dimensional Heisenberg groups.

Explicit Ricci solitons constructed on groups of real unitriangular matrices.

Demonstrated convergence of arbitrary diagonal metrics to the solitons.

Abstract

We consider Ricci flow on two classes of nilpotent Lie groups that generalize the three-dimensional Heisenberg group: the higher-dimensional classical Heisenberg groups, and the groups of real unitriangular matrices. Each group is known to admit a Ricci soliton, but we construct them \textit{explicitly} on each group. In the first case, this is done using Lott's blowdown method, whereby we demonstrate convergence of arbitrary diagonal metrics to the solitons. In the second case, which is more complicated, we obtain the solitons using a suitable ansatz.