Thin quantum-layer conjecture under bounded geometry and integrable Gauss curvature

Let $\varSigma$ be a surface satisfying the hypotheses on the embedded quantum layer, and let $B_\infty$ denote the supremum of the norm of its second fundamental form. Assume the Gauss curvature satisfies the stated integrability condition. Thin quantum-layer conjecture. There exists $\alpha=\alpha(B_\infty)>0$, depending only on the supremum of the norm of the second fundamental form, such that for every $a\in(0,\alpha)$, the discrete spectrum of the quantum layer of width $2a$ over $\varSigma$ is non-empty. The paper explains that this is proved under several additional geometric assumptions, but not in full generality; it is formulated as the remaining thin-layer problem after the known thick-layer results.