  The experimental techniques have evolved to a stage where various examples of
nanostructures with non-trivial shapes have been synthesized, turning the
dynamics of a constrained particle and the link with geometry into a realistic
and important topic of research. Some decades ago, a formalism to deduce a
meaningful Hamiltonian for the confinement was devised, showing that a
geometry-induced potential (GIP) acts upon the dynamics. In this work we study
the problem of prescribed GIP for curves and surfaces in Euclidean space
$\mathbb{R}^3$, i.e., how to find a curved region with a potential given {\it a
priori}. The problem for curves is easily solved by integrating Frenet
equations, while the problem for surfaces involves a non-linear 2nd order
partial differential equation (PDE). Here, we explore the GIP for surfaces
invariant by a 1-parameter group of isometries of $\mathbb{R}^3$, which turns
the PDE into an ordinary differential equation (ODE) and leads to cylindrical,
revolution, and helicoidal surfaces. Helicoidal surfaces are particularly
important, since they are natural candidates to establish a link between
chirality and the GIP. Finally, for the family of helicoidal minimal surfaces,
we prove the existence of geometry-induced bound and localized states and the
possibility of controlling the change in the distribution of the probability
density when the surface is subjected to an extra charge.
