SKT HYPERBOLIC AND GAUDUCHON HYPERBOLIC COMPACT COMPLEX MANIFOLDS

We introduce two notions of hyperbolicity for not necessarily K & auml;hler even balanced n-dimensional compact complex manifolds X. The first, called SKT hyperbolicity, generalises Gromov's K & auml;hler hyperbolicity by means of SKT metrics. The second, called 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 L2 harmonic spaces on the universal cover of an SKT hyperbolic manifold.