Soliton almost K\"{a}hler structures on 6-dimensional nilmanifolds for the symplectic curvature flow

TL;DR

This paper classifies certain symplectic nilpotent Lie algebras and demonstrates that specific metrics on them form soliton almost Kähler structures, advancing understanding of geometric flows on nilmanifolds.

Contribution

It provides a complete classification of symplectic two- and three-step nilpotent Lie algebras with minimal compatible metrics and shows these metrics form soliton almost Kähler structures.

Findings

Classification of symplectic nilpotent Lie algebras with minimal compatible metrics

Explicit computation of the Chern-Ricci operator for these algebras

Identification of soliton almost Kähler structures on classified nilmanifolds

Abstract

The aim of this paper is to study self-similar solutions to the symplectic cuvature flow on 6-dimensional nilmanifolds. For this purpose, we focus our attention in the family of symplectic Two- and Three-step nilpotent Lie algebras admitting a "minimal compatible metric" and we give a complete classification of these algebras together with their respective metric. Such classification is given by using our generalization of Nikolayevsky's nice basis criterium (arXiv:1301.4949), which will be repeated here in the context of canonical compatible metrics for geometric structures on nilmanifolds, for the convenience of the reader. By computing the Chern-Ricci operator $P$ in each case, we show that the above distinguished metrics define a soliton almost K\"{a}hler structure. Many illustrative examples are carefully developed.