Finite entropy vs finite energy

Abstract

Probability measures with either finite Monge–Ampère energy or finite entropy have played a central role in recent developments in Kähler geometry. In this note we make a systematic study of quasi-plurisubharmonic potentials whose Monge–Ampère measures have finite entropy. We show that these potentials belong to the finite energy class ${\mathcal{E}}^{\frac{n}{n-1}}$, where $n$ denotes the complex dimension, and provide examples showing that this critical exponent is sharp. Our proof relies on refined Moser–Trudinger inequalities for quasi-plurisubharmonic functions.