Nearly K\"ahler six-manifolds with two-torus symmetry

TL;DR

This paper studies nearly K"ahler six-manifolds with two-torus symmetry, analyzing their geometric structure via the multi-moment map, and provides a local construction method from three-dimensional data, exemplified on the Heisenberg group.

Contribution

It establishes the eigenfunction property of the multi-moment map and offers a local inverse construction for nearly K"ahler six-manifolds from 3D data.

Findings

The multi-moment map is an eigenfunction of the Laplace operator.

The $T^2$-action is free on level sets at regular values.

A local construction method for nearly K"ahler manifolds is developed.

Abstract

We consider nearly K\"ahler 6-manifolds with effective 2-torus symmetry. The multi-moment map for the $T^2$-action becomes an eigenfunction of the Laplace operator. At regular values, we prove the $T^2$-action is necessarily free on the level sets and determines the geometry of three-dimensional quotients. An inverse construction is given locally producing nearly K\"ahler six-manifolds from three-dimensional data. This is illustrated for structures on the Heisenberg group.