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

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.