The bending energy of a surface is the square of the $L^2$ norm of its second fundamental form. It arises naturally in plate and shell theories. Notably, a “very thin” non-Euclidean plate will assume a configuration that minimizes its bending energy among all its isometric immersions in $R^3$. A natural question arises: given the intrinsic geometry of a surface, bound from below its extrinsic bending energy. In this lecture I will report on two lower bounds, one associated with the distribution of curvature and another associated with a quantity which can be interpreted as a Burgers vector. Notably, the first bound detects concentrations of curvature. Applications to cones and E-cones will be demonstrated, as well as analysis of surfaces endowed with a geometry of an edge-dislocation (corresponding to a curvature dipole).
This is a joint work with Cy Maor and David Padilla-Garza.