  In this paper we provide new necessary and sufficient conditions for the
existence of K\"ahler-Einstein metrics on small deformations of a Fano
K\"ahler-Einstein manifold. We also show that the Weil-Petersson metric can be
approximated by the Ricci curvatures of the canonical $L^2$ metrics on the
direct image bundles. In addition, we describe the plurisubharmonicity of the
energy functional of harmonic maps on the Kuranishi space of the deformation of
compact K\"ahler-Einstein manifolds of general type.
