The Möbius energy, defined for closed curves embedded in Rn, is decomposed into three parts. It is called the Möbius energy since it is invariant under Möbius transformations. An alternative proof of the Möbius invariance is given using the decomposition. Furthermore the invariance of each part of decomposition is discussed.
"A decomposition theorem of the Möbius energy I: Decomposition and Möbius invariance." Kodai Math. J. 37 (3) 737 - 754, October 2014. https://doi.org/10.2996/kmj/1414674619