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Two optimization problems for the Loewner energy

Yilin Wang

TL;DR

This work studies two optimization problems for the Loewner energy, linking conformal welding on the Riemann sphere to the geometry of hyperbolic and Anti-de Sitter spaces. By exploiting two equivalent expressions of the Loewner energy—one via the driving function and another via the universal Liouville action—the authors show that minimizing energy under constraints on curves or weldings yields dual geometric structures: a pleated space-like plane boundary in $\operatorname{AdS}^3$ for the curve problem and a pleated surface boundary in $\mathbb{H}^3$ for the welding problem. The curve-optimization problem admits a unique minimizer whose welding is piecewise Möbius, while the welding-optimization problem yields piecewise circular weldings with a $C^{1,1}$ boundary curve; both problems reveal a striking symmetry between the two spaces and their boundary data. These results connect Loewner energy, universal Liouville action, and Weil–Petersson theory to concrete geometric realizations on the boundaries of $\mathbb{H}^3$ and $\operatorname{AdS}^3$, and open questions about expressing energy purely in terms of welding/AdS data and about possible Wick-rotation-type correspondences.

Abstract

A Jordan curve on the Riemann sphere can be encoded by its conformal welding, a circle homeomorphism. The Loewner energy measures how far a Jordan curve is away from being a circle, or equivalently, how far its welding homeomorphism is away from being a Möbius transformation. We consider two optimizing problems for the Loewner energy, one under the constraint for the curves to pass through $n$ given points on the Riemann sphere, which is the conformal boundary of hyperbolic $3$-space $\mathbb H^3$; the other under the constraint for $n$ given points on the circle to be welded to another $n$ given points of the circle. The latter problem can be viewed as optimizing positive curves on the boundary of AdS$^3$ space passing through $n$ prescribed points. We observe that the answers to the two problems exhibit interesting symmetries: optimizing the Jordan curve in $\partial_\infty \mathbb H^3$ gives rise to a welding homeomorphism that is the boundary of a pleated plane in AdS$^3$, whereas optimizing the positive curve in $\partial_\infty\!\operatorname{AdS}^3$ gives rise to a Jordan curve that is the boundary of a pleated plane in $\mathbb H^3$.

Two optimization problems for the Loewner energy

TL;DR

This work studies two optimization problems for the Loewner energy, linking conformal welding on the Riemann sphere to the geometry of hyperbolic and Anti-de Sitter spaces. By exploiting two equivalent expressions of the Loewner energy—one via the driving function and another via the universal Liouville action—the authors show that minimizing energy under constraints on curves or weldings yields dual geometric structures: a pleated space-like plane boundary in for the curve problem and a pleated surface boundary in for the welding problem. The curve-optimization problem admits a unique minimizer whose welding is piecewise Möbius, while the welding-optimization problem yields piecewise circular weldings with a boundary curve; both problems reveal a striking symmetry between the two spaces and their boundary data. These results connect Loewner energy, universal Liouville action, and Weil–Petersson theory to concrete geometric realizations on the boundaries of and , and open questions about expressing energy purely in terms of welding/AdS data and about possible Wick-rotation-type correspondences.

Abstract

A Jordan curve on the Riemann sphere can be encoded by its conformal welding, a circle homeomorphism. The Loewner energy measures how far a Jordan curve is away from being a circle, or equivalently, how far its welding homeomorphism is away from being a Möbius transformation. We consider two optimizing problems for the Loewner energy, one under the constraint for the curves to pass through given points on the Riemann sphere, which is the conformal boundary of hyperbolic -space ; the other under the constraint for given points on the circle to be welded to another given points of the circle. The latter problem can be viewed as optimizing positive curves on the boundary of AdS space passing through prescribed points. We observe that the answers to the two problems exhibit interesting symmetries: optimizing the Jordan curve in gives rise to a welding homeomorphism that is the boundary of a pleated plane in AdS, whereas optimizing the positive curve in gives rise to a Jordan curve that is the boundary of a pleated plane in .
Paper Structure (9 sections, 12 theorems, 49 equations, 3 figures)

This paper contains 9 sections, 12 theorems, 49 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Hyperbolic and Anti-de Sitter $3$-spaces
  3. Hyperbolic $3$-space
  4. Anti-de Sitter $3$-space
  5. Optimizing the Loewner energy
  6. Definition of the Loewner energy
  7. Optimizing the curve
  8. Optimizing the welding
  9. Comments and open questions

Key Result

Theorem 1.1

Let $z_1, \cdots, z_n$ be $n$ distinct points on $\hat{\mathbb{C}}$, $\tau$ be a Jordan curve passing through these points. There exists a unique Jordan curve $\gamma$ minimizing the Loewner energy among all curves that are homotopic to $\tau$ relatively to $z_1, \cdots, z_n$. The welding homeomorph

Figures (3)

  • Figure 1: Illustration of a pleated plane in $\operatorname{AdS}^3$ with $C^1$ boundary where $n = 4$. In this case, the four points on $\partial_\infty\!\operatorname{AdS}^3$ are co-planar.
  • Figure 2: Example of two non-homotopic curves passing through the same four points in the same order.
  • Figure 3: Conformal maps in the proof of Lemma \ref{['lem:pw_circular']}.

Theorems & Definitions (26)

  • Theorem 1.1: Optimizing the curve MRW1MRW2
  • Theorem 1.2: Optimizing the welding
  • Remark 2.1
  • Lemma 2.2: See Bonsante_Seppi
  • Lemma 2.3
  • Example 2.4
  • Lemma 2.5
  • proof
  • Remark 3.1
  • Definition 3.2
  • ...and 16 more