Dirichlet and Neumann eigenvalue problems on CR manifolds

We study the properties of Carnot–Carathéodory spaces attached to a strictly pseudoconvex CR manifold $M$, in a neighborhood of each point $x in M$, versus the pseudohermitian geometry of $M$ arising from a fixed positively oriented contact form $theta$ on $M$. The weak Dirichlet problem for the sublaplacian $Delta_b$ on $(M, theta)$ is solved on domains $Omega subset M$ supporting the Poincaré inequality. The solution to Neumann problem for the sublaplacian $Delta_b$ on a $C^{1,1}$ connected $(varepsilon , delta)$-domain $Omega subset mathbf{G}$ in a Carnot group (due to Danielli et al. in: Memoirs of American Mathematical Society 2006) is revisited for domains in a CR manifold. As an application we prove discreetness of the Dirichlet and Neumann spectra of $Delta_b$ on $Omega subset M$ in a Carnot–Carthéodory complete pseudohermitian manifold $(M,theta )$..