Finite entropy vs finite energy
Eleonora Di NezzaÉcole Polytechnique, Palaiseau, France, and Sorbonne Université, Paris, France
Vincent GuedjUniversité de Toulouse, France
Chinh H. LuUniversité Paris-Saclay, France
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 , where 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.
Cite this article
Eleonora Di Nezza, Vincent Guedj, Chinh H. Lu, Finite entropy vs finite energy. Comment. Math. Helv. 96 (2021), no. 2, pp. 389–419DOI 10.4171/CMH/515