The $$ L^2 $$ L 2 -gradient of decomposed Möbius energies

Abstract

The Mobius energy is an example of a knot energy, so-called since it is invariant under Mobius transformations. It has been shown that it is defined on the fractional Sobolev space \( H^{ \frac{3}{2} } \cap W^{1, \infty } \), where it can be decomposed into three parts, each of which retains the Mobius invariance. These results hold not only for knots in three-dimensional Euclidean space but also for closed curves embedded in n -dimensional Euclidean space. It has already been obtained that the variational formulae of the decomposed energies for both curves and their variations are in the same fractional Sobolev space. A formal integration by parts implies that the formulae extend to linear forms for variationals in \( L^2 \) if the curve is in \( H^3 \). Such a linear form is called the \( L^2 \)-gradient. In this paper, we confirm this expectation, and give an explicit expression of the \( L^2 \)-gradient for the decomposed Mobius energies including all lower-order terms.