Description
Let $X$ be a compact Riemann surface. For a positive line bundle $L\to X$, a continuous Hermitian metric $e^{-\phi}h_\ast$ on $L$, and a nonpolar compact subset $K\subseteq X$, one can construct an equilibrium metric from the envelope of subharmonic weight functions bounded above by $\phi$ on $K$. The approximation of equilibrium metrics by sequences of singular metrics has been studied in a variety of contexts, including complex geometry, potential theory, dynamical systems, and probability theory.
In this talk, I will introduce two functions, called entropy and free energy, associated with semipositive singular Hermitian metrics on the canonical bundle of a Riemann surface. These quantities are defined using an extension of the notion of complete harmonic metrics on cyclic Higgs bundles, whose existence and uniqueness were established by Li--Mochizuki, to complete Hermitian metrics associated with semipositive singular metrics on the canonical bundle that are not necessarily induced by holomorphic $r$-differentials. I will then discuss my ongoing research aimed at quantitatively establishing the principles of entropy increase and free energy decrease in various situations where an equilibrium metric is approximated by a sequence of singular metrics.