A SYMMETRIC CRITICALITY PRINCIPLE FOR O'HARA'S ENERGIES (Analysis on Shapes of Solutions to Partial Differential Equations)

Abstract

In geometric knot theory, one tries to examine knots, i.e. embedded curves in R3 by using energy functionals. These so-called knot energies shall determine whether and to what extent a curve is knotted. We focus on the O'Hara energies, which are a family of smooth knot energies, and try to evaluate their energy landscape. Using a modified version of Palais' symmetric criticality principle, we show the existence of at least two distinct critical knots for O'Hara's energies. These critical knots are smooth.