The Stability of Generalized Ricci Solitons

Abstract

In [12,28], 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 the dynamical stability and the linear stability are equivalent on a steady gradient generalized Ricci soliton (g, H, f ). This generalizes the results in [15,22,25,31,33].

Bismut Ricci flat manifolds with symmetries

“…3] for more information. We recall here that for a BRF pair (g, H) the scalar curvature Scal g and the norm of H, which are related by the identity Scal g = 1 4 ||H|| 2 , are constant on M, see [18,Lemma 2.24]. BRF pairs are fixed points of the generalized Ricci flow, a geometric flow introduced in [8,19] in the context of renormalization group flows of two-dimensional nonlinear sigma models.…”