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We construct in this article a class of closed semi-bounded quadratic formson the space of square integrable functions over a smooth Riemannian manifoldwith smooth boundary. Each of these quadratic forms specifies a semi-boundedself-adjoint extension of the Laplace-Beltrami operator. These quadratic formsare based on the Lagrange boundary form on the manifold and a family of domainsparametrized by a suitable class of unitary operators on the boundary that willbe called admissible. The corresponding quadratic forms are semi-bounded belowand closable. Finally, the representing operators correspond to semi-boundedself-adjoint extensions of the Laplace-Beltrami operator. This family ofextensions is compared with results existing in the literature and variousexamples and applications are discussed.

Self-adjoint extensions of the Laplace-Beltrami operator and unitaries at the boundary