# The Ricci pinching functional on solvmanifolds II

## TL;DR

This paper investigates whether solvsolitons are the global maxima of the Ricci pinching functional on solvable Lie groups, extending previous results to more general cases involving nilradicals and metric restrictions.

## Contribution

It proves that solvsolitons are the unique global maxima of the Ricci pinching functional in broader classes of solvable Lie groups, including those with nilradicals of codimension one.

## Key findings

- Solvsolitons are global maxima for the Ricci pinching functional in new cases.
- The results extend previous work to groups with nilradicals of codimension one.
- The study includes cases where the nilradical is abelian and metrics are restricted.

## Abstract

It is natural to ask whether solvsolitons are global maxima for the Ricci pinching functional F:=scal^2/|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 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) s has a nilradical of codimension-one; 2) the nilradical n of s is abelian and the functional F is restricted to the set of metrics such that a is orthogonal to n, where a is the orthogonal complement of n with respect to the solvsoliton.