Piecewise geodesic Jordan curves II: Loewner energy, projective structures, and accessory parameters

TL;DR

The paper establishes that in each relative isotopy class of Jordan curves on the sphere through $n\ge3$ prescribed points, there exists a unique Loewner-energy minimizer, which is a smooth piecewise geodesic whose arcs are hyperbolic geodesics in the complement of the other arcs. This minimizer induces a marked projective structure $Z_\gamma$ with $\mathrm{PSL}(2,\mathbb{R})$-holonomy, and the Schwarzian comparing $Z_\gamma$ to the trivial structure has simple poles at the punctures whose residues equal the Wirtinger derivatives $\partial_{z_k} I^L(\gamma)/2$, linking energy variation to accessory parameters in a Polyakov-type fashion. The authors further show that the welding parametrization of these curves yields a $C^1$ diffeomorphism between the space of smooth piecewise Möbius weldings and the Teichmüller space $T_{0,n}$, and prove that the Fuchsian projective structures relate to the piecewise geodesic constructions by differentiable $\pi$-grafting. Collectively, these results provide a canonical bridge between Loewner energy minimizers, projective structures with $PSL(2,\mathbb{R})$-holonomy, and grafting dynamics on the $n$-punctured sphere, with implications for the analytic Langlands program via real opers. $

Abstract

In this paper we consider Jordan curves on the Riemann sphere passing through $n \ge 3$ given points. We show that in each relative isotopy class of such curves, there exists a unique curve that minimizes the Loewner energy. These curves have the property that each arc between two consecutive points is a hyperbolic geodesic in the domain bounded by the other arcs. This geodesic property lets us define a complex projective structure whose holonomy lies in $\mathrm{PSL}(2,\mathbb{R})$. We show that the quadratic differential comparing this projective structure to the trivial projective structure on the sphere has simple poles whose residues (accessory parameters) are given by the Wirtinger derivatives of the minimal Loewner energy. This is reminiscent of Polyakov's conjecture for Fuchsian projective structures, proven by Takhtajan and Zograf. Finally, we show that the projective structures we obtain are related to Fuchsian projective structures through $π$-grafting.

Figures (7)

• Figure 1: The marking of $\pi_1$ and a set of generators.
• Figure 2: Illustration of the uniformizing conformal maps associated with a standard geodesic pair in $(\mathbb{D}; \mathrm{e}^{\mathfrak{i} \theta}, \mathrm{e}^{-\mathfrak{i} \theta}; 0)$. It was computed in MRW1 (see Figure 1 there) that $A = (\pi/2) \cos (\theta)$, $B = \sin (\theta) + (\pi /2 -\theta) \cos (\theta)$, $C = - 2A - B$, $D = 2A - B$.
• Figure 3: The covering space picture. Left: The values of $h_\alpha$ on the arcs $\ell_\beta$ are written in red. Arcs with the same color are equivalent under the deck transformations. Right: The rooted tree $\mathcal{T}_\alpha$ is drawn with dashed lines.
• Figure 4: $C^1$ smooth geodesic pairs in $\mathbb{D}$ and $\mathbb{H}$.
• Figure 5: Grafting by $\theta$ along $\mathfrak{i} \mathbb{R}_+$.

Theorems & Definitions (82)

• Proposition 2.1
• Example 2.2
• Lemma 2.3
• proof
• Definition 2.4
• Remark 2.5
• Lemma 2.6: Additivity of Loewner energy
• Definition 2.7
• Theorem 2.8: RW
• Theorem 2.9: W2