We study the natural functional $F=\frac {\operatorname {scal}^2}{|\operatorname {Ric}|^2}$ on the space of all non-flat left-invariant metrics on all solvable Lie groups of a given dimension $n$. As an application of properties of the beta operator, we obtain that solvsolitons are the only global maxima of $F$ restricted to the set of all left-invariant metrics on a given unimodular solvable Lie group, and beyond the unimodular case, we obtain the same result for almost-abelian Lie groups. Many other aspects of the behavior of $F$ are clarified.

中文翻译：

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Ricci捏在功能块上起作用

我们研究自然可函数$ F = \ frac {\ operatorname {scal} ^ 2} {| \ operatorname {Ric} | ^ 2} $在a的所有可解Lie组的所有非平坦左不变度量的空间上给定维度$ n $。作为beta算符性质的一种应用，我们获得了孤子是唯一限制于给定单模可解Lie组上所有左不变度量集的$ F $的全局最大值，并且在单模情况下，我们获得了对于几乎是阿拉伯人的李氏团体，结果也是一样。$ F $的行为的许多其他方面都得到了澄清。