Almost soliton duality

TL;DR

This paper explores the properties and dualities of gradient Ricci almost solitons, revealing their conformal relationships, uniqueness conditions, and completeness criteria, especially in Kähler settings with special potentials.

Contribution

It demonstrates that almost solitons are conformally related to other almost solitons with negated soliton functions and investigates their uniqueness and completeness properties.

Findings

Almost solitons are conformal to other almost solitons with negated soliton functions.

The paper establishes conditions for the uniqueness of almost solitons.

Completeness of the target metric is analyzed in Kähler cases with special potentials.

Abstract

Gradient Ricci almost solitons were introduced by Pigola, Rigoli, Rimoldi and Setti. They are defined as solitons except that the metric coefficient is required to be a smooth function rather than a constant. It is shown that any almost soliton is conformal to another almost soliton having a soliton function which is minus the original one. Uniqueness, and the case where both the source and target are solitons, are studied. Completeness of the target metric is also examined in the case where the source is K\"ahler and admits a special K\"ahler-Ricci potential in the sense given by Derdzinski and Maschler.