Regularity theory for the Möbius energy

The Möbius energy, defined 1991 by O'Hara, is the most prominent example of a knot energy. In this text we will focus on the regularity of local minimizers (within a prescribed knot class) whose arc-length parametrization was shown to be $C^{1,1}$ by Freedman, He, and Wang. Later on, He improved this result to $C^\infty$ regularity. In this text we will briefly outline the main ideas of these two steps which require completely different approaches involving techniques from geometry and analysis. Moreover we explain how to rigorously derive the first variation of the Möbius energy and fix a gap in He's treatise.