The Ricci pinching functional on solvmanifolds

TL;DR

This paper investigates the Ricci pinching functional on solvmanifolds, establishing that solvsolitons are the unique global maxima among left-invariant metrics, with extensions to almost-abelian groups.

Contribution

It proves that solvsolitons uniquely maximize the Ricci pinching functional on unimodular solvable Lie groups and extends this result to almost-abelian groups.

Findings

Solvsolitons are the only global maxima of F on unimodular solvable Lie groups.

The same maximality result holds for almost-abelian Lie groups.

The paper clarifies various behaviors of the Ricci pinching functional.

Abstract

We study the natural functional F=scal^2/|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.