Loop Loewner Energy and its root invariance

Abstract: Loop Loewner Energy is a conformally invariant quantity measuring how far a simple loop L embedded in the Riemann sphere differs from a circle. It is defined through the Loewner differential equation, which contributes a lot to the understanding of univalent functions. The caveat of the Loewner loop energy is that it a priori depends on the choice of a point x on L called the root of L.  Recently with Steffen Rohde, we prove that the energy is actually independent of the root for a regular enough loop.