A deterministic approach to Loewner-energy minimizers

TL;DR

This paper provides a deterministic approach to identify curves minimizing Loewner energy, deriving explicit formulas and proving uniqueness, and reveals a universal algebraic curve underlying all such minimizers.

Contribution

It introduces a purely deterministic methodology to analyze Loewner energy minimizers, deriving explicit formulas and establishing a universal curve for welding minimizers.

Findings

Explicit energy formulas for minimizers

Existence and uniqueness of minimizers proven

Identification of a universal algebraic curve

Abstract

We study two minimization questions: the nature of curves $\gamma \subset \mathbb{H}$ which minimize the Loewner energy among all curves from 0 to a fixed $z_0 \in \mathbb{H}$, and the nature of $\gamma$ which minimize the Loewner energy among all curves that weld a given pair $x<0 <y$. The former question was partially studied by Yilin Wang, who used SLE techniques to calculate the minimal energy and show it is uniquely attained. We revisit the question using a purely deterministic methodology, and re-derive the energy formula and also obtain further results, such as an explicit computation of the driving function. Our approach also yields existence and uniqueness of minimizers for the welding question, as well as an explicit energy formula and explicit driving function. In addition, we show both families have a "universality" property; for the welding minimizers this means that there is a single, explicit algebraic curve $\Gamma$ such that truncations of $\Gamma$ or its reflection $-\overline{\Gamma}$ in the imaginary axis generate all welding minimizers up to scaling. While Wang noted her minimizer is SLE$_0(-8)$, we show the welding minimizers are SLE$_0(-4,-4)$. Our results also show sharpness of a case of the driver-curve regularity theorem of Carto Wong.