Abstract

This survey focuses on geometric curvature functionals, that is, geometrically defined self-avoidance energies for curves, surfaces or more general $k$-dimensional sets in $\mathbb{R}^d$. Previous investigations of the authors and collaborators concentrated on the regularising effects of such energies, with a priori estimates in the regime above scale-invariance that allowed for compactness and variational applications for knotted curves and surfaces under topological restrictions. We briefly describe the impact of geometric curvature energies on geometric knot theory. Currently, various attempts are being made to obtain a deeper understanding of the energy landscape of these highly singular and nonlinear nonlocal interaction energies. Moreover, a regularity theory for critical points is being developed in the setting of fractional Sobolev spaces. We describe some of these current trends and present a list of open problems.

Reference

In: P. Reiter, S. Blatt, and A. Schikorra (Eds.). New Directions in Geometric and Applied Knot Theory. Berlin, Boston: De Gruyter (pp. 8–35).