Tian’s Properness Conjectures:An Introduction to Kähler Geometry

Abstract

This manuscript served as lecture notes for a minicourse in the 2016 Southern California Geometric Analysis Seminar Winter School. The goal is to give a quick introduction to Kähler 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 Kähler–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.