Singularity Formation and Stability in Ricci Flow on Compact Four-Manifolds with Positive Isotropic Curvature

This monograph investigates the rigorous analytic structure of singularity formation and geometric stability within the Ricci flow on compact four-dimensional Riemannian manifolds satisfying the Positive Isotropic Curvature (PIC) condition. We establish that the PIC condition is strictly preserved under the flow, serving as a robust barrier against the formation of non-generic singularities. By analyzing the evolution of the curvature operator R acting on the complexified bundle 2 TCM, we classify the possible Type I and Type II singularity models, demonstrating that ancient -solutions in this setting are necessarily quotients of S4 or S3 R. Furthermore, we detail the surgery procedure required to extend the flow beyond singular times, providing a comprehensive proof of the topological classification of such manifolds as connected sums of spherical space forms and S3 S1. The stability of the flow is analyzed through the spectral properties of the linearized Lichnerowicz Laplacian, confirming the dynamical isolation of the singular set.