Dynamical stability of Pluriclosed and Generalized Ricci solitons

TL;DR

This work analyzes the dynamical stability of pluriclosed flow and generalized Ricci flow on compact complex manifolds. It develops a second variation operator $\overline{N}_f$ and uses an affine generalized slice to relate linear stability to a local maximality of the generalized Einstein–Hilbert functional $\lambda$, yielding dynamical stability results when infinitesimal solitonic deformations are integrable. In the pluriclosed setting, the authors derive a specialized second variation formula involving $\xi$ and $\eta$ and establish stability results for both fixed and varying complex structures, including the Bismut-flat case where exponential convergence is obtained. When the first Chern class vanishes, the pluriclosed flow is shown to converge (up to automorphisms) to a steady pluriclosed soliton, with rigidity results within fixed Aeppli classes. Overall, the paper connects variational, gauge, and complex-geometric methods to establish local stability and convergence of generalized and pluriclosed flows near solitons.

Abstract

In this work, we discuss the stability of the pluriclosed flow and generalized Ricci flow. We proved that if the second variation of generalized Einstein--Hilbert functional is nonpositive and the infinitesimal deformations are integrable, the flow is dynamically stable. Moreover, we prove that the pluriclosed steady solitons are dynamically stable when the first Chern class vanishes.

Theorems & Definitions (100)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Proposition 1.5
• Corollary 1.6
• Theorem 1.7
• Theorem 1.8
• Definition 2.1
• Definition 2.2