The Stability of Generalized Ricci Solitons

In Garcia-Fernandez and Streets (Generalized Ricci flow, volume 76 of university lecture series, American Mathematical Society, Providence, 2021) and Oliynyk et al. (Nucl Phys B 739(3):441–458, 2006), it was shown that the generalized Ricci flow is the gradient flow of a functional λ generalizing Perelman’s λ functional for Ricci flow. In this work, we further computed the second variation formula and proved that a Bismut-flat, Einstein manifold is linearly stable under some curvature assumptions. In the last part of this paper, I proved that dynamical stability and linear stability are equivalent on a steady gradient generalized Ricci soliton ( g , H , f ). This generalizes the results in Haslhofer and Müller (Math Ann 360(1–2):547–553, 2014), Kröncke (Stability of Einstein Manifolds, 2014, Commun Anal Geom 28(2):351–394, 2020), Raffero and Vezzoni (On the dynamical behaviour of the generalized Ricci flow, 2020) and Sesum (Duke Math J 133(1):1–26, 2006).