Singularity Formation and Perelman Entropy Functionals in Ricci Flow on Non-Compact Kähler Manifolds

This monograph investigates the asymptotic behavior of the Kähler-Ricci flow on complete, non-compact Kähler manifolds, focusing on the formation of finite-time singularities. We extend Perelman's W-entropy functional to the non-compact setting by introducing a weighted functional space adapted to the decay of the Kähler potential. We establish a rigorous monotonicity formula for the extended W-functional under suitable curvature and potential growth assumptions. Furthermore, we analyze the blow-up limits of Type I singularities, proving that they converge, in the Cheeger-Gromov sense, to non-flat gradient shrinking Kähler-Ricci solitons. Special attention is given to the stability of the "blowdown soliton" and the role of the potential function in controlling the topology of the singular limit. Our results provide a variational characterization of singularity models in the non-compact regime, generalizing the Hamilton-Tian conjecture to open manifolds.