  Let $M$ be a closed complex submanifold in ${\mathbb C}^N$ with the complete
K\"ahler metric induced by the Euclidean metric. Several finiteness theorems on
the $L^p$ Bergman space of holomorphic sections of a given Hermitian line
bundle $L$ over $M$ and the associated $L^2$ cohomology groups are obtained.
Some infiniteness theorems are also given in order to test the accuracy of
finiteness theorems. As applications we obtain some rigidity results concerning
growth of curvatures.
