Let $M$ be a complete Riemannian manifold and $-\Delta$ denote the Laplace-Beltrami operator on $M$. We can prove that $(-\Delta f, g) = (\nabla f, \nabla g) = (f, -\Delta g)$, when $f, g \in C^\infty_0(M)$. My question is, when one extends the Laplace-Beltrami operator as a self-adjoint operator, what is the domain of the extension?
Edit: As Jack Lee points out, here we are thinking of $-\Delta$ as an unbounded operator on $L^2(M)$.
Take a look at this reference from Strichartz http://www.sciencedirect.com/science/article/pii/0022123683900903
The operator is essentially selfadjoint on its natural domain. The domain is, as the one commenter noted, $H^{2}(M)$.