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

Abstract

Let $\left(X,\omega \right)$ be a compact connected Kähler manifold and denote by $\left({\mathcal{ℰ}}^p,d_p\right)$ the metric completion of the space of Kähler potentials ${\mathcal{\mathscr{H}}}_{\omega }$ 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 ${\mathcal{ℰ}}^p$ 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 $\left({\mathcal{ℰ}}^2,d_2\right)$. 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 ${\mathcal{ℰ}}^2$. This gives the first concrete result about the long-time convergence of this flow on general Kähler manifolds, partially confirming a conjecture of Donaldson. 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. Finally, when a cscK metric exists in ${\mathcal{\mathscr{H}}}_{\omega }$, our results imply that the weak Calabi flow $d_1$–converges to such a metric.