SKT Hyperbolic and Gauduchon Hyperbolic Compact Complex Manifolds

We introduce two notions of hyperbolicity for not necessarily K\"ahler even balanced $n$-dimensional compact complex manifolds $X$. The first, called {\it SKT hyperbolicity}, generalises Gromov's K\"ahler hyperbolicity by means of SKT metrics. The second, called {\it Gauduchon hyperbolicity} by means of Gauduchon metrics. Our first main result in this paper asserts that every SKT hyperbolic $X$ is also Kobayashi/Brody hyperbolic and every Gauduchon hyperbolic $X$ is divisorially hyperbolic. The second main result is to prove a vanishing theorem for the $L^2$ harmonic spaces on the universal cover of a SKT hyperbolic manifold.