Concerning a conjecture of Taketomi-Tamaru

TL;DR

The paper investigates left-invariant Ricci solitons on 2-step nilpotent Lie groups with non-exceptional type $(p,q)$, showing that generically the orbits of $\mathbb{R}^{>0}\times Aut_0$ acting on $GL(n)/O(n)$ are congruent even when a soliton metric exists, thereby providing a counterexample to the local version of the Taketomi-Tamaru conjecture. The authors develop a framework based on the $j$-map to the Lie algebra $\mathfrak{so}(q)^p$, study the induced $GL(q)\times GL(p)$ action, and apply moment-map stability criteria to identify closed orbits corresponding to nilsoliton metrics. They prove that for non-exceptional types there is a Zariski open set of algebras with minimal derivation algebras that yield nilsolitons and congruent yet non-transitive orbits, culminating in an explicit 9-dimensional example of type $(4,5)$ with a closed orbit and small stabilizer that serves as the counterexample. This work highlights that local orbit geometry alone cannot fully determine soliton existence and motivates a global-action perspective on the Taketomi-Tamaru conjecture.

Abstract

We study the setting of 2-step nilpotent Lie groups in the particular case that its type (p,q) is not exceptional. We demonstrate that, generically, the orbits of $\mathbb R^{>0}\times Aut_0$ in $GL(n)/O(n)$ are congruent even when a Ricci soliton metric does exists. In doing so, we provide a counterexample to the local version of a conjecture of Taketomi-Tamaru.

Theorems & Definitions (16)

• Definition 2.1
• Theorem 2.2
• Lemma 2.3
• Remark 2.4
• proof : Proof of the lemma
• Theorem 3.1
• Proposition 3.2
• Proposition 3.3
• proof
• Theorem 4.1