Boltzmann's Entropy and K\"ahler-Ricci Solitons

We study a Boltzmann's type entropy functional (which appeared in existing literature) defined on K\"ahler metrics of a fixed K\"ahler class. The critical points of this functional are gradient K\"ahler-Ricci solitons, and the functional was known to be monotonically increasing along the K\"ahler-Ricci flow in the canonical class. In this article, we derive and analyze the second variation formula for this entropy functional, and show that all gradient K\"ahler-Ricci solitons are stable with respect to this entropy functional. Furthermore, using this result, we give a new proof that gradient shrinking K\"ahler-Ricci solitons are stable with respect to the Perelman's entropy in a fixed K\"ahler class.