  This article is an expanded version of the plenary talk given by Evans
Harrell at QMath98, a meeting in Prague, June 1998. We consider Laplace
operators and Schr\"odinger operators with potentials containing curvature on
certain regions of nontrivial topology, especially closed curves, annular
domains, and shells. Dirichlet boundary conditions are imposed on any
boundaries. Under suitable assumptions we prove that the fundamental eigenvalue
is maximized when the geometry is round. We also comment on the use of
coordinate transformations for these operators and mention some open problems.
