Available online xxxx Communicated by Gang Tian MSC: 53C55 32W20 32U05We introduce different Finsler metrics on the space of smooth Kähler potentials that will induce a natural geometry on various finite energy classes Eχ(X, ω). Motivated by questions raised by R. Berman, V. Guedj and Y. Rubinstein, we characterize the underlying topology of these spaces in terms of convergence in energy and give applications of our results to existence of Kähler-Einstein metrics on Fano manifolds.

Well-known conjectures of Tian predict that existence of canonical Kähler 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 Kähler-Einstein metrics, with or without automorphisms, as well as for Kähler-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 Kenergy in the Finsler metric completion. Finally, our results also resolve Tian's conjecture on the Moser-Trudinger inequality for Fano manifolds with Kähler-Einstein metrics.

We establish the monotonicity property for the mass of non-pluripolar products on compact Kähler manifolds, and we initiate the study of complex Monge-Ampère type equations with prescribed singularity type. Using the variational method of Berman-Boucksom-Guedj-Zeriahi we prove existence and uniqueness of solutions with small unbounded locus. We give applications to Kähler-Einstein metrics with prescribed singularity, and we show that the log-concavity property holds for nonpluripolar products with small unbounded locus.Without the non-zero mass condition X θ n φ > 0 this characterization cannot hold (see Remark 3.3). The equivalence between (i) and (iii) in the above theorem shows that P θ [u] is the same potential for any u ∈ E(X, θ, φ), and equals to P θ [φ]. Given this and the inclusion E(X, θ, φ) ⊂ E(X, θ, P θ [φ]), one is tempted to consider only potentials φ in the image of the operator ψ → P θ [ψ], when studying the classes of relative full mass E(X, θ, φ). These potentials seemingly play the same role as V θ , the potential with minimal V * K,θ is supported on K, we thus haveand the desired inequality holds in this case. If M := M φ (K) ≥ 1, then by (12) we have φ ≤ M −1 V * K,φ + (1 − M −1 )φ ≤ φ + 1, and by definition of the relative capacity we can write:

Suppose (X, ω) is a compact Kähler manifold. Following Mabuchi, the space of smooth Kähler potentials H can be endowed with a Riemannian structure, which induces an infinite dimensional path length metric space (H, d). We prove that the metric completion of (H, d) can be identified with (E 2 (X, ω),d), and this latter space is a complete non-positively curved geodesic metric space. In obtaining this result, we will rely on envelope techniques which allow for a treatment in a very general context. Profiting from this, we will characterize the pairs of potentials in PSH(X, ω) that can be connected by weak geodesics and we will also give a characterization of E(X, ω) in this context.

Let (X, ω) be a compact connected Kähler manifold and denote by (E p , d p ) the metric completion of the space of Kähler potentials H ω with respect to the L p -type path length metric d p . First, we show that the natural analytic extension of the (twisted) Mabuchi K-energy to E p is a d p -lsc functional that is convex along finite energy geodesics. Second, following the program of J. Streets, we use this to study the asymptotics of the weak (twisted) Calabi flow inside the CAT(0) metric space (E 2 , d 2 ). This flow exists for all times and coincides with the usual smooth (twisted) Calabi flow whenever the latter exists. We show that the weak (twisted) Calabi flow either diverges with respect to the d 2 -metric or it d 1 -converges to some minimizer of the K-energy inside E 2 . This gives the first concrete result about the long time convergence of this flow on general Kähler manifolds, partially confirming a conjecture of Donaldson. Finally, we investigate the possibility of constructing destabilizing geodesic rays asymptotic to diverging weak (twisted) Calabi trajectories, and give a result in the case when the twisting form is Kähler. If the twisting form is only smooth, we reduce this problem to a conjecture on the regularity of minimizers of the K-energy on E 1 , known to hold in case of Fano manifolds.