A deterministic proof of Loewner energy reversibility via local reversals

We give a new proof of the orientation reversibility of chordal Loewner energy by reversing the orientation of a chord in partial increments. This fact was first proved by Yilin Wang (arXiv:1601.05297) using the reversibility of chordal Schramm-Loewner evolution (SLE) along with the interpretation of Loewner energy as the large deviation rate function of chordal SLE${}_{\kappa }$ as $\kappa \to 0$. Our method is similar in spirit to Dapeng Zhan’s proof (arXiv:0808.3649) of chordal SLE${}_{\kappa }$ reversibility for $\kappa \in (0,4]$, though it is purely deterministic. As a key step in our proof, we establish that a minimimal energy chord among those passing through a fixed finite set of points is a piecewise hyperbolic geodesic.