On the existence of balanced metrics on six-manifolds of cohomogeneity one
Izar Alonso, Francesca Salvatore
TL;DR
The paper addresses the existence of balanced non-Kähler SU(3)-structures on six-dimensional manifolds with cohomogeneity-one group actions. By reducing to the principal part M^{princ} and classifying possible (𝔤,𝔨) pairs consistent with a G-invariant SU(3)-structure, the authors perform a case-by-case analysis and derive global nonexistence results for three of the potential types, using stability, compatibility, and extension obstructions. The main outcome, encapsulated in Theorem A and Theorem B, is that, except for a specific open compact-type case, there are no G-invariant balanced non-Kähler SU(3)-structures on simply connected six-manifolds with cohomogeneity-one actions; in particular, S^3×S^3 does not admit such a structure under a cohomogeneity-one symmetry. This narrows the landscape of invariant solutions to the Hull–Strominger system in this setting and highlights the role of global extendability versus local invariants in constrained geometric structures.
Abstract
We consider balanced metrics on complex manifolds with holomorphically trivial canonical bundle, most commonly known as balanced $\rm{SU}(n)$-structures. Such structures are of interest for both Hermitian geometry and string theory, since they provide the ideal setting for the Hull-Strominger system. In this paper, we provide a non-existence result for balanced non-Kähler $\rm{SU}(3)$-structures which are invariant under a cohomogeneity one action on simply connected six-manifolds.