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

We consider the Möbius energy defined for a closed curve in ${\mathbb{R}}^n$: $$\mathcal{M}(\text{boldmath }f)={\iint }_{(\mathbb{R}/\mathcal{L}\mathbb{Z}{)}^2}(\frac{1}{\Vert \text{boldmath }f(s_1)-\text{boldmath }f(s_2){\Vert }_{{\mathbb{R}}^n}^2}-\frac{1}{\mathcal{D}(\text{boldmath }f(s_1),\text{boldmath }f(s_2){)}^2})d{s}_{1}d{s}_2.$$ Here $\mathcal{L}$ is the length of closed curve, $s_i$'s are arc-length parameters, and $\mathcal{D}$ is the distance along the curve.

In this talk we show that the energy can be decomposed into three parts: $$\mathcal{M}(\text{boldmath }f)={\mathcal{M}}_1(\text{boldmath }f)+{\mathcal{M}}_2(\text{boldmath }f)+4.$$ The first one is an analogue of Gagliardo semi-norm of ${\text{boldmath }f}^{\prime }$ in the fractional Sobolev space $H^{1/2}$. This implies the natural domain of $\mathcal{M}$ is $H^{3/2}\cap H^{1,\mathrm{\infty }}$, 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.