Regularity theory for knot energies
Seminar Room 1, Newton Institute
In the past two decades, the introduction of several knot-based geometric functionals has greatly contributed to the field of geometric curvature energies.
The general aim is investigating geometric properties of a given knotted curve in order to gain information on its knot type. More precisely, the original idea was to search a "nicely shaped" representative in a given knot class having strands being widely apart. This led to modeling self-avoiding functionals, so-called knot energies, that blow up on embedded curves converging to a curve with a self-intersection.
Due to the singularities which guarantee the self-repulsion property all these functionals lead to interesting analytical problems which in many cases almost naturally involve fractional Sobolev spaces.
In this talk we consider stationary points of knot energies. To this end we compute the Euler-Lagrange equation and derive higher regularity via a bootstrapping process.