Tian’s properness conjectures and Finsler geometry of the space of Kähler metrics

Abstract:

Well-known conjectures of Tian predict that the existence of canonical Kähler metrics should be equivalent to various notions of properness of Mabuchi’s K-energy functional. First, we provide counterexamples to Tian’s first conjecture in the presence of continuous automorphisms. Second, we resolve Tian’s second conjecture, confirming the Moser–Trudinger inequality for Fano manifolds. 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 replacements for Tian’s first conjecture in the setting of smooth and singular Kähler–Einstein metrics, with or without automorphisms, as well as for Kähler–Ricci solitons. Moreover, we reduce both Tian’s original first 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.