K\"ahler-Ricci flows coming out of metric spaces

TL;DR

This paper investigates conditions under which positive currents on compact K"ahler manifolds induce metric structures that are limits of K"ahler manifolds or flows, extending results to Alexandrov surfaces in dimension one.

Contribution

It provides new criteria linking positive currents to metric limits of K"ahler manifolds and extends K"ahler-Ricci flow initial data characterization to Alexandrov surfaces.

Findings

Positive currents can induce metric structures as Gromov-Hausdorff limits.

Conditions are identified for these structures to arise from K"ahler-Ricci flows.

Any Alexandrov surface with bounded curvature and no cusp can serve as initial data for the flow.

Abstract

Given a compact K\"ahler manifold $X$ and a closed, positive $(1,1)$-current $T$ on $X$, we find sufficient conditions for $T$ to induce a metric structure $(X,d_T)$ which is the Gromov-Hausdorff limit of compact K\"ahler manifolds either in a "static" way or at time zero of smooth K\"ahler-Ricci flows. In dimension $1$ we extend works of T. Richard and M. Simon, showing that any oriented compact Alexandrov surface with bounded integral curvature and without cusp is the initial datum of a K\"ahler-Ricci flow.