Let (M,g) be compact Riemannian manifold (possibly $\partial M\neq\emptyset)$
Now I have read, that "the laplace-beltrami operator is a positive definite operator".
I have shown, if M is a closed manifold or if we consider dirichlet boundary conditions, $\Delta:=-\nabla^2$ is positive definite, i.e. $(\Delta f,f)_{L^2}\geq 0$.
Which setting does the author mean by: "the laplace-beltrami operator is a positive definite operator"?? Is $\Delta$ always positive definite independent of the boundary conditions?
Thank you!