Tian's properness conjectures: an introduction to Kahler geometry

TL;DR

This paper introduces the resolution of Tian's properness conjectures in Kahler geometry, linking analytic conditions to the existence of canonical metrics and establishing key inequalities, thus advancing understanding in complex differential geometry.

Contribution

It presents the proof of Tian's properness conjectures, providing a new analytic framework for Kahler--Einstein metrics and related geometric inequalities.

Findings

Resolution of Tian's properness conjectures

Analytic characterization of Kahler--Einstein metrics

Establishment of strong Moser--Trudinger inequalities

Abstract

This manuscript served as lecture notes for a mini-course in the 2016 Southern California Geometric Analysis Seminar Winter School. The goal is to give a quick introduction to Kahler geometry by describing the recent resolution of Tian's three influential properness conjectures in joint work with T. Darvas. These results---inspired by and analogous to work on the Yamabe problem in conformal geometry---give an analytic characterization for the existence of Kahler--Einstein metrics and other important canonical metrics in complex geometry, as well as strong borderline Sobolev type inequalities referred to as the (strong) Moser--Trudinger inequalities.