  This paper is about the influence of Geometry on the qualitative behaviour of
solutions of quasilinear PDEs on Riemannian manifolds. Motivated by examples
arising, among others, from the theory of submanifolds, in particular by the
study of entire graphs with prescribed mean curvature, we consider classes of
coercive differential inequalities of the form $$ \mathrm{div}\left(
\frac{\varphi(|\nabla u|)}{|\nabla u|} \nabla u\right) \ge b(x)f(u) l(|\nabla
u|) \qquad \text{(respectively, $\le$ or $=$)} $$ on domains of a manifold $M$,
for suitable $\varphi,b,f,l$, with emphasis on mean curvature type operators.
We investigate the validity of strong maximum principles, compact support
principles and Liouville type theorems; in particular, the goal is to identify
sharp thresholds, involving curvatures or volume growth of geodesic balls in
$M$, to guarantee the above properties under appropriate Keller-Osserman type
conditions, and to discuss the geometric reasons behind the existence of such
thresholds. The paper also aims to give a unified view of recent results in the
literature. The bridge with Geometry is realized by studying the validity of
weak and strong maximum principles at infinity, in the spirit of Omori-Yau's
Hessian and Laplacian principles and subsequent improvements.
