Energy shifts resulting from surface curvature of quantum nanostructures

We use differential forms to derive the quantum-mechanical potential induced by surface curvature. Two surfaces are modeled with the classical Monge representation and the resulting modifications to the potential- and kinetic-energy operators in the Schr\"odinger equation are obtained. We find that the ground-state energy shift resulting from the dot's surface curvature is of sufficient order to be observable in quantum nanostructures.