Entropy for Monge-Ampère Measures in the Prescribed Singularities Setting

Abstract

In this note, we generalize the notion of entropy for potentials in a relatively full Monge-Ampère mass ℰ(𝑋, 𝜃, 𝜙), for a model potential 𝜙. We then investigate stability properties of this condition with respect to blow-ups and perturbation of the cohomology class. We also prove a Moser-Trudinger type inequality with general weight, and we show that functions with finite entropy lie in a relative energy class ℰ↑(𝑛/(𝑛−1))(𝑋, 𝜃, 𝜙) (provided 𝑛 > 1), while they have the same singularities of 𝜙 when 𝑛 = 1.