Essential Self-Adjointness of Perturbed Biharmonic Operators via Conformally Transformed Metrics

Abstract

We give sufficient conditions for the essential self-adjointness of perturbed biharmonic operators acting on sections of a Hermitian vector bundle over a Riemannian manifold with additional assumptions, such as lower semi-bounded Ricci curvature or bounded sectional curvature. In the case of lower semi-bounded Ricci curvature, we formulate our results in terms of the completeness of the metric that is conformal to the original one, via a conformal factor that depends on a minorant of the perturbing potential V. In the bounded sectional curvature situation, we are able to relax the growth condition on the minorant of V imposed in an earlier article. In this context, our growth condition on the minorant of V is consistent with the literature on the self-adjointness of perturbed biharmonic operators on \(\mathbb {R}^{n}\).