Stability of Ricci Solitons under Generalized Geometric Flows on Fano Manifolds

This monograph investigates the linear and dynamic stability of gradient shrinking Ricci solitons on compact Fano manifolds under a class of generalized geometric flows. We extend the canonical stability analysis of Perelman's -entropy to the context of the modified Kähler-Ricci flow driven by a holomorphic vector field. By establishing a second variation formula for the generalized entropy functional, we derive a precise spectral condition on the weighted Lichnerowicz Laplacian governing the linear stability of the soliton. Furthermore, we prove that strict linear stability implies local dynamic stability, ensuring exponential convergence of the flow to the soliton structure modulo diffeomorphisms. The analysis integrates techniques from spectral theory, geometric analysis of parabolic flows, and the theory of K-stability, providing a rigorous framework for understanding the asymptotic behavior of Fano manifolds admitting non-trivial soliton structures.