Tian's properness conjectures and Finsler geometry of the space of Kahler metrics

TL;DR

This paper investigates Tian's properness conjectures relating to Kahler metrics, providing counterexamples with automorphisms, introducing a Finsler metric approach, and establishing new optimal conjectures and results in Kahler geometry.

Contribution

It introduces a novel Finsler metric approach to properness, constructs counterexamples to Tian's original conjecture with automorphisms, and formulates refined conjectures with proven results.

Findings

Counterexamples to Tian's original conjecture with automorphisms.

A new Finsler metric framework for properness.

Resolution of Tian's conjecture on Moser-Trudinger inequality.

Abstract

Well-known conjectures of Tian predict that existence of canonical Kahler metrics should be equivalent to various notions of properness of Mabuchi's K-energy functional. In some instances this has been verified, especially under restrictive assumptions on the automorphism group. We provide counterexamples to the original conjecture in the presence of continuous automorphisms. The construction hinges upon an alternative approach to properness that uses in an essential way the metric completion with respect to a Finsler metric and its quotients with respect to group actions. This approach also allows us to formulate and prove new optimal versions of Tian's conjecture in the setting of smooth and singular Kahler-Einstein metrics, with or without automorphisms, as well as for Kahler-Ricci solitons. Moreover, we reduce both Tian's original conjecture (in the absence of automorphisms) and our modification of it (in the presence of automorphisms) in the general case of constant scalar curvature metrics to a conjecture on regularity of minimizers of the K-energy in the Finsler metric completion. Finally, our results also resolve Tian's conjecture on the Moser-Trudinger inequality for Fano manifolds with Kahler-Einstein metrics.