New inhomogeneous Bismut Hermitian-Einstein threefolds with large symmetry group

We study compact non-Kähler Bismut Hermitian-Einstein threefolds whose connected full group of holomorphic isometries acts with cohomogeneity one. We prove that the Lie algebra of this group is isomorphic to $\mathbb{R}^3\oplus\mathfrak{su}(2)$ or $\mathbb{R}^2\oplus\mathfrak{su}(2)$, unless the Bismut connection is flat. We then focus on the first case, where we determine the possible singular isotropy algebras and reduce the Bismut Hermitian-Einstein condition to a system of ordinary differential equations. We obtain explicit solutions satisfying suitable smoothness conditions, that yield families of compact and complete non-compact non-Kähler Bismut Hermitian-Einstein threefolds with non-constant scalar curvature and non-zero Bismut curvature, including examples on $S^3\times S^3$ and $S^3\times S^1\times\mathbb{R}^2$, respectively. We also investigate the topology of the compact examples and determine the asymptotic scalar curvature of the non-compact ones. Moreover, we prove the existence of a smooth deformation of a Bismut-flat Samelson structure on $S^3\times S^3$ through non-Kähler Bismut Hermitian-Einstein structures that are not Bismut-flat. The regularity of the constructed compact examples is also investigated.