Energy spectrum of a constrained quantum particle and the Willmore energy of the constraining surface

In this paper, we establish geometric and topological upper bounds on the first energy level gap of a particle confined to move on a compact surface in $3$-space. Our main contribution is proving that the first gap in the energy spectrum of a confined particle (a physical property) is bounded above by the Willmore energy of the confining surface (a geometric property). Furthermore, we demonstrate that the only surfaces that permit a confined particle with a stationary and uniformly distributed wave function are surfaces with constant skew curvature.