Equivalent Descriptions of the Loewner Energy

Abstract

Loewner's equation provides a way to encode a simply connected domain or equivalently its uniformizing conformal map via a real-valued driving function of its boundary. The first main result of the present paper is that the Dirichlet energy of this driving function (also known as the Loewner energy) is equal to the Dirichlet energy of the log-derivative of the (appropriately defined) uniformizing conformal map. This description of the Loewner energy then enables to tie direct links with regularized determinants and Teichmüller theory: We show that for smooth simple loops, the Loewner energy can be expressed in terms of the zeta-regularized determinants of a certain Neumann jump operator. We also show that the family of finite Loewner energy loops coincides with the Weil-Petersson class of quasicircles, and that the Loewner energy equals to a multiple of the universal Liouville action introduced by Takhtajan and Teo, which is a Kähler potential for the Weil-Petersson metric on the Weil-Petersson Teichmüller space.

Figures (7)

• Figure 1: We often choose the half-planes to be $\mathbb{H}$ and the lower half-plane $\mathbb{H}^*$ as the image of $h_1$ and $h_2$ to fit into the Loewner chain setting. However, it is clear that the last two expressions of the equality in Theorem \ref{['thm_chord_int_as_thm']} is invariant under transformations $z \mapsto az +b$, for $a \in \mathbb{C}^*$ and $b\in \mathbb{C}$.
• Figure 2: The infinite capacity curve $\overline{\gamma}$ is the completion of $\gamma$ by adding the conformal geodesic $\overline{\gamma} \setminus \gamma = h_T^{-1} (\mathbb{R}_-)$ connecting $\gamma_T$ to $\infty$ in $\Sigma \setminus \gamma[0,T]$.
• Figure 3: Illustration of the definition of loop driving function $W : \mathbb{R} \to \mathbb{R}$ and the capacity reparameterization $t\mapsto \Gamma(t)$ of $\gamma$, where $0< s_2<s_0<s_1$ correspond to the capacities $-\infty < t_2 < 0 < t_1$.
• Figure 4: Conformal mappings in the proof of Lemma \ref{['lem_arc_identity']} where $\varphi_t$ is defined in the complement of $\Gamma[-\infty,t]$ and $h$ in the complement of $\Gamma[-\infty ,T]$. Both of them map the tips to tips.
• Figure 5: Conformal mappings in the proof of Theorem \ref{['thm_loop_identity']}. We define $\varphi_n (z) = (h_n)^{-1} (C z + h_n(0^+))$ on $\mathbb{H}$ and $\varphi_n (z)= (h_n)^{-1} (C' z + h_n(0^-))$, where $C$ and $C'$ are chosen such that $\varphi_n$ fixes $0,1$ and $\infty$.

Theorems & Definitions (51)

• Theorem 1.1
• Theorem 1.2: see Theorem \ref{['thm_loop_identity']}
• Theorem 1.3
• Theorem 1.4
• Proposition 2.1
• Theorem 1: see LoopEnergycarto-wong
• Corollary 2.2
• proof
• Theorem 2: Kellogg's theorem, see e.g. GM2005Harmonic Thm. II.4.3
• Lemma 3.1: Extension of Stokes' formula