Notices of the International Consortium of Chinese Mathematicians

Volume 11 (2023)

Number 1

Auxiliary Monge–Ampère equations in geometric analysis

Pages: 98 – 135

DOI: https://dx.doi.org/10.4310/ICCM.2023.v11.n1.a10

Authors

Bin Guo (Department of Mathematics and Computer Science, Rutgers University, Newark, New Jersey, U.S.A.)

Duong H. Phong (Department of Mathematics, Columbia University, New York, N.Y., U.S.A.)

Abstract

This is an introduction to a particular class of auxiliary complex Monge–Ampère equations which have been instrumental in $L^\infty$ estimates for fully non-linear equations and various questions in complex geometry. The essential comparison inequalities are reviewed and shown to apply in many contexts. Adapted to symplectic geometry, with the auxiliary equation given now by a real Monge–Ampère equation, the method gives an improvement of an earlier theorem of Tosatti–Weinkove–Yau, reducing Donaldson’s conjecture on the Calabi–Yau equation with a taming symplectic form from an exponential bound to an $L^1$ bound.

Published 11 July 2023