Spectrum of the Laplacian in waveguide shaped surfaces

Let $-{\mathrm{\Delta }}_{\mathcal{S}}$ be the Laplace operator in $\mathcal{S}\subset {\mathbb{R}}^3$ on a waveguide shaped surfaces, i.e., $\mathcal{S}$ is built by translating a closed curve in a constant direction along an unbounded spatial curve. Under the condition that the tangent vector of the reference curve admits a finite limit at infinity, we find the essential spectrum of $-{\mathrm{\Delta }}_{\mathcal{S}}$ and discuss conditions under which discrete eigenvalues emerge. Furthermore, we analyze the Laplacian in the case of a broken sheared waveguide shaped surface.