Pluriclosed Flow and the Geometrization of Complex Surfaces

Abstract

We recall fundamental aspects of the pluriclosed flow equation and survey various existence and convergence results, and the various analytic techniques used to establish them. Building on this, we formulate a precise conjectural description of the long time behavior of the flow on complex surfaces. This suggests an attendant geometrization conjecture which has implications for the topology of complex surfaces and the classification of generalized Kähler structures.