Equivariant isotopy of unknots to round circles (Q1896635)

The Möbius group \(\text{Möb} (\mathbb{S}^3)\) is the (10)-dimensional Lie group of two components generated by inversions in 2-sphere. The authors prove that for every compact subgroup \(G\) of \(\text{Möb}(\mathbb{S}^3)\), if \(\gamma \subset \mathbb{S}^3\) is a smooth unknotted \(G\)-invariant simple closed curve, then there exists a smooth family of \(G\)-invariant simple closed curves \(\gamma_t\), \(0 \leq t \leq 1\), with \(\gamma_0 = \gamma\) and \(\gamma_1\) a round circle. This interesting result links with the study of the existence of a Möbius invariant energy functional defined on the set of smooth simple closed curves in the 3-sphere (with values in the positive real numbers), attaining its minimum precisely on round circles.