On the Ground State of Quantum Layers

The problem is from mesoscopic physics: let p : Σ → R3 be an embedded surface in R3, we assume that (1) Σ is orientable, complete, but non-compact; (2) Σ is not totally geodesic; (3) Σ is asymptotically flat in the sense that the second fundamental form goes to zero at infinity. On can build a quantum layer Ω over such a surface Σ as follows: as a differentiable manifold, Ω = Σ × [−a, a] for some positive number a. Let ~ N be the unit normal vector of Σ in R3. Define p̃ : Ω → R