A rigidity result for axisymmetric toric Ricci solitons

TL;DR

This paper proves that certain axisymmetric Ricci solitons are rigid against non-axisymmetric perturbations, leading to a classification of Einstein metrics and Ricci solitons under specific conditions.

Contribution

It establishes a rigidity result for axisymmetric toric Ricci solitons, showing they cannot be non-axisymmetrically perturbed unless conformally flat, and provides explicit classifications.

Findings

Non-axisymmetric perturbations are not admitted except in conformally flat cases

Explicit descriptions of Einstein metrics are derived

Classification of Ricci solitons under volume-collapsing conditions

Abstract

We examine a non-axisymmetric perturbation of a family of axisymmetric toric Einstein manifolds and Ricci solitons studied in Firester-Tsiamis (2024). We establish a rigidity result stating that these axisymmetric Ricci solitons do not admit constant-angle non-axisymmetric perturbations except for conformally flat cases. For these new cases, our result leads to an explicit description of the Einstein metrics and a classification of the Ricci solitons under a volume-collapsing ansatz.