In this paper, we establish maximum principles for weakly 1-coercive operators L on complete, non-compact Riemannian manifolds M. In particular, we search for conditions under which one can guarantee that solutions u of differential equations of the form \(L(u)\geq f(u)\) satisfy \(f(u)\leq 0\) on M. The case of weakly p-coercive operators with \(p>1\) , including the p-Laplacian and in particular the Laplace-Beltrami operator for \(p=2\) , has been considered in our recent paper [2]. As a consequence of the main results, we infer comparison principles for that kind of operators. Furthermore, we apply them to geometric situations dealing with the mean curvature operator, which is a typical weakly 1-coercive operator. We first consider the case of \(\mathcal C^1\) operators L acting on functions u of class \(\mathcal C^2\) and, in the last section of the paper, we show how our results can be extended to the case of less regular operators L acting on functions u which are just continuous and locally \(W^{1,1}\) regular.