The Ricci pinching functional on solvmanifolds II

Abstract

It is natural to ask whether solvsolitons are global maxima for the Riccipinching functional F := scal^2/ |Ric|^2 on the set of all left-invariant metrics on a given solvableLie group S, as it is to ask whether they are the only global maxima. A positive answer toboth questions was given in a recent paper by the same authors when the Lie algebra s ofS is either unimodular or has a codimension-one abelian ideal. In the present paper, weprove that this also holds in the following two cases: 1) s has a nilradical of codimension-one; 2) the nilradical n of s is abelian and the functional F is restricted to the set ofmetrics such that a is orthogonal to n, where s = a + n is the orthogonal decomposition with respectto the solvsoliton.