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This paper is about the influence of Geometry on the qualitative behaviour ofsolutions of quasilinear PDEs on Riemannian manifolds. Motivated by examplesarising, among others, from the theory of submanifolds, in particular by thestudy of entire graphs with prescribed mean curvature, we consider classes ofcoercive differential inequalities of the form $$ \mathrm{div}\left(\frac{\varphi(|\nabla u|)}{|\nabla u|} \nabla u\right) \ge b(x)f(u) l(|\nablau|) \qquad \text{(respectively, $\le$ or $=$)} $$ 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 supportprinciples and Liouville type theorems; in particular, the goal is to identifysharp thresholds, involving curvatures or volume growth of geodesic balls in$M$, to guarantee the above properties under appropriate Keller-Osserman typeconditions, and to discuss the geometric reasons behind the existence of suchthresholds. The paper also aims to give a unified view of recent results in theliterature. The bridge with Geometry is realized by studying the validity ofweak and strong maximum principles at infinity, in the spirit of Omori-Yau'sHessian and Laplacian principles and subsequent improvements.

On the interplay among maximum principles, compact support principles and Keller-Osserman conditions on manifolds