On O'hara knot energies I: Regularity for critical knots

TL;DR

This paper develops a regularity theory for extremal knots of scale-invariant O'hara energies, extending understanding beyond the M"obius energy by connecting to fractional harmonic maps and overcoming invariance limitations.

Contribution

It introduces a novel approach to regularity for O'hara energies by reinterpreting them as nonlinear nonlocal energies on tangent vectors, linking to fractional harmonic map theory.

Findings

Proves regularity of minimizers and critical points of O'hara energies.

Establishes a connection between knot energies and fractional harmonic maps.

Extends regularity results beyond M"obius invariant energies.

Abstract

We develop a regularity theory for extremal knots of scale invariant knot energies defined by J. O'hara in 1991. This class contains as a special case the M\"obius energy. For the M\"obius energy, due to the celebrated work of Freedman, He, and Wang, we have a relatively good understanding. Their approch is crucially based on the invariance of the M\"obius energy under M\"obius transforms, which fails for all the other O'hara energies. We overcome this difficulty by re-interpreting the scale invariant O'hara knot energies as a nonlinear, nonlocal $L^p$-energy acting on the unit tangent of the knot parametrization. This allows us to draw a connection to the theory of (fractional) harmonic maps into spheres. Using this connection we are able to adapt the regularity theory for degenerate fractional harmonic maps in the critical dimension to prove regularity for minimizers and critical knots of the scale-invariant O'hara knot energies.