  Using the stress energy tensor, we establish some monotonicity formulae for
vector bundle-valued p-forms satisfying the conservation law, provided that the
base Riemannian (resp. K\"ahler) manifolds poss some real (resp. complex)
p-exhaustion functions. Vanishing theorems follow immediately from the
monotonicity formulae under suitable growth conditions on the energy of the
p-forms. As an application, we establish a monotonicity formula for the Ricci
form of a K\"ahler manifold of constant scalar curvature and then get a growth
condition to derive the Ricci flatness of the K\"ahler manifold. In particular,
when the curvature does not change sign, the K\"ahler manifold is isometrically
biholomorphic to C^m. Another application is to deduce the monotonicity
formulae for volumes of minimal submanifolds in some outer spaces with suitable
exhaustion functions. In this way, we recapture the classical volume
monotonicity formulae of minimal submanifolds in Euclidean spaces. We also
apply the vanishing theorems to Bernstein type problem of submanifolds in
Euclidean spaces with parallel mean curvature. In particular, we may obtain
Bernstein type results for minimal submanifolds, especially for minimal real
K\"ahler submanifolds under weaker conditions.
