Existence of ground states for quantum layers over asymptotically flat surfaces

The conjecture asks whether every nonflat quantum layer over an asymptotically flat surface with integrable Gauss curvature has a ground state, under the stated minimal assumptions. It was explicitly formulated in the 2011 literature; the full claim was not proved there.

Known results

• Ruled outside a compact set: asymptotic flatness and nonflatness imply a ground state (2007).
• Nonpositive total Gauss curvature: the asymptotically planar case has a ground state (Duclos–Exner–Krejčiřík, reported in 2004).
• Additional sufficient hypotheses include weak $\kappa$-parabolicity, nonnegative Gauss curvature, and parabolicity for sufficiently thin layers (2011).
• Convex asymptotically flat graphs give positive-curvature examples with ground states, but not the general positive-total-curvature case (2007).

Search through August 2026

No retrieved source reports a complete proof, counterexample, error correction, or verification of a claimed solution. A 2022 approach treats stronger geometric settings, including asymptotically cut-locus-planar surfaces, but does not settle this conjecture.

Current status (as of August 2026): The conjecture remains open; ground-state existence is settled only under additional geometric assumptions or in special classes of surfaces.