A deterministic approach to Loewner-energy minimizers

Abstract

We study two minimization questions: the nature of curves $γ\subset \mathbb{H}$ which minimize the Loewner energy among all curves from 0 to a fixed $z_0 \in \mathbb{H}$, and the nature of $γ$ 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 $Γ$ such that truncations of $Γ$ or its reflection $-\overlineΓ$ 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.

Figures (11)

• Figure 1: The "mapping down" function $g_t$ "unzipping" the curve $\gamma$.
• Figure 2: The downwards and upwards Loewner flows for $\gamma = \gamma[0,T]$.
• Figure 3: The limiting curve $\gamma_0$ for the EMP curves and its reflection $\gamma_0^*$ across $\partial \mathbb{D}$ form a boundary geodesic pair, in the sense that $\gamma_0$ is the hyperbolic geodesic from 0 to 1 in its component of $\mathbb{H} {\backslash} \gamma_0^*$, and $\gamma_0^*$ is a hyperbolic geodesic from 1 to $\infty$ in its component of $\mathbb{H} {\backslash} \gamma_0$.
• Figure 4: The universal curve $\Gamma$ for the welding minimizers with ratios $0 <r = -x/y <1$. Starting from closest to the imaginary axis and moving counterclockwise, the four points show where to truncate for EMW curves with welding endpoint ratios $r=0.75, 0.5, 0.25$ and $0.01$, respectively. The complex square of $\Gamma$ is a circle tangent at the origin to $\mathbb{R}$.
• Figure 5: EMW and EMP curves in orange and blue, respectively, that weld the same black points $x_0<0<y_0$ to their base. Note the increase in argument of the tip when moving from the Wang to the welding minimizer, which is expected from the monotonicity of \ref{['Eq:Energy1']}.

Theorems & Definitions (51)

• Theorem A
• Corollary A.1: Corollary \ref{['Cor:BoundaryGeodesic']}
• Corollary A.2: Corollary \ref{['Cor:WongSharp']}
• Theorem B
• Lemma C: Lemma \ref{['Lemma:UpwardsSLE']}
• Theorem D: Theorem \ref{['Thm:EMWWang']}
• Theorem E: Theorem \ref{['Lemma:Inf98']}
• Definition 2.1
• Proposition 2.1
• Proposition 2.2