Weakly holomorphic homogeneous regular manifolds

We introduce a class of complex manifolds which we call weakly holomorphic homogeneous regular manifolds (wHHR) manifolds. As the name suggests, this class contains the so-called holomorphic homogeneous regular manifolds but also other classes of complex manifolds such as two dimensional finite type domains and simply connected Kähler manifolds with pinched negative sectional curvature. For wHHR Stein manifolds we prove that the Bergman and Kobayashi metrics are biLipschitz equivalent.