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arXiv · 2610.29514

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

Andrea Bisterzo·Salvatore Lincastri·Giona Veronelli
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Abstract

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 $p\in (1,+\infty)$.

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Andrea Bisterzo, Salvatore Lincastri, Giona Veronelli. 2026-10-21. $L^p$-Liouville properties and realizations of Schrödinger operators on incomplete Riemannian manifolds. https://arxiv.org/abs/2610.29514

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