A new proof of Perelman's scalar curvature and diameter estimates for the K\"ahler-Ricci flow on Fano manifolds

In this note, we give a new proof for Perelman's scalar curvature and diameter estimates for the K\"ahler-Ricci flow on Fano manifolds. The proof relies on a new Harnack estimate for a special family of functions in space-time. Our new approach initiates the work in \cite{JST23a} for general finite time solutions of the K\"ahler-Ricci flow.