On the linear stability of expanding Ricci solitons

TL;DR

This paper investigates the linear stability of expanding Ricci solitons with bounded curvature, including solvsolitons, demonstrating their stability after extension by a Gaussian soliton using advanced analytical techniques.

Contribution

It advances the understanding of Ricci soliton stability by proving linear stability for a broad class of expanding solitons, extending previous results to include Gaussian extensions.

Findings

Expanding Ricci solitons with bounded curvature are linearly stable after Gaussian extension.

The stability results support the conjecture that all solvsolitons are linearly stable.

The techniques generalize previous methods by Guenther, Isenberg, and Knopf.

Abstract

In previous work, the authors studied the linear stability of algebraic Ricci solitons on simply connected solvable Lie groups (solvsolitons), which are stationary solutions of a certain normalization of Ricci flow. Many examples were shown to be linearly stable, leading to the conjecture that all solvsolitons are linearly stable. This paper makes progress towards that conjecture, showing that expanding Ricci solitons with bounded curvature (including solvsolitons) are linearly stable after extension by a Gaussian soliton. As in the previous work, the dynamical stability follows from a generalization of the techniques of Guenther, Isenberg, and Knopf.