The pluriclosed flow for $T^2$-invariant Vaisman metrics on the Kodaira-Thurston surface

TL;DR

This paper investigates the behavior of $T^2$-invariant Vaisman metrics on the Kodaira-Thurston surface under the pluriclosed flow, showing the flow preserves the Vaisman condition even with non-constant scalar curvature.

Contribution

It characterizes $T^2$-invariant Vaisman metrics on the Kodaira-Thurston surface and demonstrates the pluriclosed flow preserves the Vaisman condition in this setting.

Findings

Vaisman metrics with non-constant scalar curvature exist on the Kodaira-Thurston surface.

The pluriclosed flow preserves the Vaisman condition for $T^2$-invariant metrics.

Extension of previous results to non-constant scalar curvature case.

Abstract

In this note we study $T^2$-invariant pluriclosed metrics on the Kodaira-Thurston surface. We obtain a characterization of $T^2$-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 $T^2$-invariant Vaisman metric, then the pluriclosed flow preserves the Vaisman condition, extending to the non-constant scalar curvature case the previous results in [6].