A deterministic proof of Loewner energy reversibility via local reversals

TL;DR

This paper provides a new deterministic proof of the reversibility of Loewner energy by incrementally reversing chord orientations, avoiding probabilistic methods and establishing geometric properties of minimal energy chords.

Contribution

It introduces a deterministic approach to prove Loewner energy reversibility, differing from prior probabilistic proofs, and characterizes minimal energy chords as piecewise hyperbolic geodesics.

Findings

Reversibility of Loewner energy proven deterministically

Minimal energy chords are piecewise hyperbolic geodesics

Method avoids reliance on SLE reversibility properties

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 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.