Convexity of the extended K-energy and the large time behavior of the weak Calabi flow

Abstract

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.

Plurisubharmonic envelopes and supersolutions

“…Proof. The idea of the proof is essentially contained in [GZ07], [BDL15]. It costs no generality to assume that u j ≤ 0 which also implies that u ≤ 0.…”

Relative K-stability for Kähler manifolds

“…extremal) metric on a projective variety implies K-stability [8,18,25,44] (resp. relative K-stability [46]).…”

Geometric Pluripotential Theory on Sasaki Manifolds

“…of the extended K-energy along the finite energy geodesic segments follows exactly as in [7][Theorem 4.7].…”