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Well-known conjectures of Tian predict that existence of canonical Kahlermetrics should be equivalent to various notions of properness of Mabuchi'sK-energy functional. In some instances this has been verified, especially underrestrictive assumptions on the automorphism group. We provide counterexamplesto the original conjecture in the presence of continuous automorphisms. Theconstruction hinges upon an alternative approach to properness that uses in anessential way the metric completion with respect to a Finsler metric and itsquotients with respect to group actions. This approach also allows us toformulate and prove new optimal versions of Tian's conjecture in the setting ofsmooth and singular Kahler-Einstein metrics, with or without automorphisms, aswell as for Kahler-Ricci solitons. Moreover, we reduce both Tian's originalconjecture (in the absence of automorphisms) and our modification of it (in thepresence of automorphisms) in the general case of constant scalar curvaturemetrics to a conjecture on regularity of minimizers of the K-energy in theFinsler metric completion. Finally, our results also resolve Tian's conjectureon the Moser-Trudinger inequality for Fano manifolds with Kahler-Einsteinmetrics.

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