Rigidity results for non-Kähler Calabi-Yau geometries on threefolds

We derive a canonical symmetry reduction associated to a compact non-Kähler Bismut-Hermitian-Einstein manifold. In real dimension $6$, the transverse geometry is conformally Kähler, and we give a complete description in terms of a single scalar PDE for the underlying Kähler structure. In the case when the soliton potential is constant, we show that that the Bott-Chern number $h_{BC}^{1,1}\ge 2$, and that equality holds if and only if the metric is Bismut-flat, and hence a quotient of either $\text{SU}(2)\times \mathbb{R}\times \mathbb{C}$ or $\text{SU}(2)\times \text{SU}(2)$.