The pluriclosed flow for T2-invariant Vaisman metrics on the Kodaira-Thurston surface

Abstract

In this note we study T2-invariant pluriclosed metrics on the Kodaira-Thurston surface. We obtain a characterization of T2-invariant Vaisman metrics, and notice that the Kodaira-Thurston surface admits Vaisman metrics with non-constant scalar curvature. Then we study the behaviour of the Vaisman condition in relation to the pluriclosed flow. As a consequence, we show that if the initial metric on the Kodaira-Thurston surface is a T2-invariant Vaisman metric, then the pluriclosed flow preserves the Vaisman condition, extending to the non-constant scalar curvature case the previous result in [6].