$L^p$-Liouville properties and realizations of Schrödinger operators on incomplete Riemannian manifolds

The aim of the paper is to highlight and standardize in a self-contained framework the connections between some properties associated with the Laplace-Beltrami operator on possibly incomplete Riemannian manifolds. We present both new and well-known characterizations of the positivity-preserving property, essential self-adjointness, and Liouville-type results for $L^2$-functions, using the latter to establish a clear hierarchy among these properties. Furthermore, we prove the stability of the essential self-adjointness of the Laplace-Beltrami operator on incomplete Riemannian manifolds under the removal of closed (possibly non-compact) subsets from a complete manifold, subject to suitable conditions on their (generalized) Hausdorff measure. With appropriate modifications, this framework is adapted throughout the paper to the general $L^p$-case for $p\in (1,+\mathrm{\infty })$.