The Stability of Generalized Ricci Solitons

TL;DR

This paper analyzes the stability of generalized Ricci solitons by deriving second variation formulas and establishing conditions under which these solitons are linearly stable, extending previous results to a broader class.

Contribution

It computes the second variation of the generalized Einstein-Hilbert functional and proves the equivalence of dynamical and linear stability for steady gradient generalized Ricci solitons.

Findings

Bismut-flat Einstein manifolds are linearly stable under certain curvature conditions.

Dynamical and linear stability are equivalent for steady gradient generalized Ricci solitons.

Abstract

In this paper, I computed the second variation formula of the generalized Einstein-Hilbert functional and prove that a Bismut-flat, Einstein manifold is linearly stable under some curvature assumption. In the last part of the paper, I prove that dynamical stability and linear stability are equivalent on a steady gradient generalized Ricci soliton $(g, H,f)$ which generalizes the result done by Kr\"oncke, Haslhofer, Sesum, Raffero, and Vezzoni.