A deterministic proof of Loewner energy reversibility via local reversals

Abstract

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 Wang (2019) 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 Zhan's proof (2008) 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 minimal energy chord among those passing through a fixed finite set of points is a piecewise hyperbolic geodesic.