Decomposition of the Möbius energy I: A decomposition theorem and Möbius invariance

We consider the Möbius energy defined for a closed curve in $ \mathbb{R}^n $: \[ \mathcal{M} ( \mbox{\boldmath $ f $} ) = \iint_{ ( \mathbb{R} / \mathcal{L} \mathbb{Z} )^2 } \left(\frac 1 { \| \mbox{\boldmath $ f $} ( s_1 ) - \mbox{\boldmath $ f$} ( s_2 ) \|_{\mathbb{R}^n }^2 } - \frac 1 { \mathscr{D} ( \mbox{\boldmath $ f $} ( s_1 ) , \mbox{\boldmath $f $} ( s_2 ) )^2 } \right) d s_1 d s_2 .\] Here $ \mathcal{L} $ is the length of closed curve, $ s_i $'s are arc-length parameters, and $ \mathscr{D} $ is the distance along the curve.

In this talk we show that the energy can be decomposed into three parts: \[\mathcal{M} ( \mbox{\boldmath $ f $} ) = \mathcal{M}_1 ( \mbox{\boldmath $ f $} ) + \mathcal{M}_2 ( \mbox{\boldmath $ f $} ) + 4 .\] The first one is an analogue of Gagliardo semi-norm of $ \mbox{\boldmath $ f $}^\prime $ in the fractional Sobolev space $ H^{1/2 } $. This implies the natural domain of $ \mathcal{M} $ is $ H^{ 3/2 } \cap H^{1,\infty } $, which was shown by Blatt. The integrand of second one has the determinant structure, which shows a cancellation of integrand.

The energy $ \mathcal{M} $ is invariant under the Möbius transformations. We discuss the Möbius invariance of each $ \mathcal{M}_i $'s.