CMX Lunch Seminar
Surfaces which minimize a squared curvature bending energy are fundamental in the theory of smooth surfaces, as well as in geometric and physical modeling. The canonical representative of such energies is the Willmore energy, measuring the total squared mean curvature of a surface. The associated Euler-Lagrange equation is a non-linear 4th order PDE which presents significant numerical challenges.
In this talk I will introduce some of the tools we developed towards effective algorithms for finding minimizers of the Willmore energy in a given conformal class, i.e., allowing only conformal deformations. Such (conformally) constrained Willmore surfaces can be understood as generalizations of non-linear splines from the univariate to the bivariate setting. Physically these model isotropic auxetic materials. Our algorithms also serve as tools for experimental mathematics in the study of extrinsic surface shape as a function of the metric and genus.
Joint work with Yousuf Soliman (Caltech), Olga Diamanti (UGraz), Albert Chern (UCSD), Felix Knöppel (TU Berlin), Ulrich Pinkall (TU Berlin)