Canonical almost-K\"ahler metrics dual to general plane-fronted wave Lorentzian metrics

TL;DR

This paper constructs extremal and second-Chern-Einstein non-K"ahler almost-K"ahler metrics that are dual to general plane-fronted wave Lorentzian metrics, extending previous work on their Riemannian counterparts.

Contribution

It introduces a method to build non-K"ahler almost-K"ahler metrics with special curvature properties dual to Lorentzian plane-fronted wave metrics.

Findings

Construction of extremal almost-K"ahler metrics.

Development of second-Chern-Einstein almost-K"ahler metrics.

Extension of duality results to non-K"ahler settings.

Abstract

In the compact setting, Aazami and Ream \cite{Aazami:2022th} proved that Riemannian metrics dual to a class of Lorentzian metrics, called (compact) general plane-fronted waves, are almost-K\"ahler. In this note, we explain how to construct extremal and second-Chern-Einstein non-K\"ahler almost-K\"ahler metrics dual to those general plane-fronted waves.