Geometric invariants and the Monge-Ampere equation in K\"ahler geometry

TL;DR

This paper surveys recent geometric inequalities and estimates related to the Monge-Ampere equation in Kähler geometry, highlighting their dependence on Yau's solution of the Calabi conjecture and exploring current research directions.

Contribution

It provides a comprehensive overview of new geometric inequalities and estimates for the Monge-Ampere equation developed by the authors and collaborators.

Findings

New geometric inequalities for the Monge-Ampere equation

Estimates dependent on Yau's solution of the Calabi conjecture

Survey of current directions in complex geometry

Abstract

This is a contribution to the special issue of Surveys in Differential Geometry celebrating the 75th birthday of Shing-Tung Yau. The bulk of the paper is devoted to a survey of some new geometric inequalities and estimates for the Monge-Ampere equation, obtained by the authors in the last few years in joint work with F. Tong, J. Song, and J. Sturm. These all depend in an essential way on Yau's solution of the Calabi conjecture, which is itself nearing its own 50th birthday. The opportunity is also taken to survey briefly many current directions in complex geometry, which he more recently pioneered.