Finite entropy vs finite energy

TL;DR

This paper explores the relationship between finite entropy and finite Monge-Ampère energy in Kähler geometry, establishing that potentials with finite entropy belong to a specific finite energy class and identifying the sharpness of this exponent.

Contribution

It systematically studies potentials with finite entropy, proves their inclusion in a critical finite energy class, and introduces refined inequalities to support these findings.

Findings

Potentials with finite entropy are in the finite energy class ${\\mathcal E}^{\frac{n}{n-1}}$.

The critical exponent for this class is sharp, as shown by examples.

Refined Moser-Trudinger inequalities are used in the proof.

Abstract

Probability measures with either finite Monge-Amp\`ere energy or finite entropy have played a central role in recent developments in K\"ahler geometry. In this note we make a systematic study of quasi-plurisubharmonic potentials whose Monge-Amp\`ere 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.