  Using Rauch's comparison theorem, we prove several monotonicity inequalities
for Riemannian submanifolds. Our main result is a general Li-Yau inequality
which is applicable in any Riemannian manifold whose sectional curvature is
bounded above (possibly positive). We show that the monotonicity inequalities
can also be used to obtain Simon type diameter bounds, Sobolev inequalities and
corresponding isoperimetric inequalities for Riemannian submanifolds with small
volume. Moreover, we infer lower diameter bounds for closed minimal
submanifolds as corollaries. All the statements are intrinsic in the sense that
no embedding of the ambient Riemannian manifold into Euclidean space is needed.
Apart from Rauch's comparison theorem, the proofs mainly rely on the first
variation formula, thus are valid for general varifolds.
