The Ricci pinching functional on solvmanifolds II

Abstract:

It is natural to ask whether solvsolitons are global maxima for the Ricci pinching functional $F\coloneq \frac {\textrm {scal}^2}{|\textrm {Ric}|^2}$ on the set of all left-invariant metrics on a given solvable Lie group $S$, as it is to ask whether they are the only global maxima. A positive answer to both questions was given in a recent paper by the same authors when the Lie algebra $\mathfrak {s}$ of $S$ is either unimodular or has a codimension-one abelian ideal. In the present paper, we prove that this also holds in the following two more general cases: 1) $\mathfrak {s}$ has a nilradical of codimension-one; 2) the nilradical $\mathfrak {n}$ of $\mathfrak {s}$ is abelian and the functional $F$ is restricted to the set of metrics such that $\mathfrak {a}\perp \mathfrak {n}$, where $\mathfrak {s}=\mathfrak {a}\oplus \mathfrak {n}$ is the orthogonal decomposition with respect to the solvsoliton.