We prove several vanishing theorems for the cohomology of balanced hyperbolic manifolds that we introduced in our previous work and for the [Formula: see text] harmonic spaces on the universal cover of these manifolds. Other results include a Hard Lefschetz-type theorem for certain compact complex balanced manifolds and the non-existence of certain [Formula: see text] currents on the universal covering space of a balanced hyperbolic manifold.

In this note, we establish a connection between SKT and balanced hyperbolicity, highlighting their relationship with Kähler hyperbolicity in the sense of Gromov. MSC code: 2020: 32Q45, 53C55, 14F25, 14F40, 53C30.