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.

Geodesic curvature formula

As many of our proofs rely on the following formula on the variation of the geodesic curvature under a Weyl-scaling, we sketch a short proof for readers’ convenience.

Lemma A.1

Let $(D, g_0)$ be a surface with smooth boundary $\gamma = \partial D$. If $\sigma \in C^{\infty } (D, \mathbb {R})$ and $g = e^{2 \sigma } g_0$, the geodesic curvature of $\partial D$ under the metric g satisfies

$$\begin{aligned} k_g = e^{-\sigma } \left( k_0 + \partial _{n_0} \sigma \right) , \end{aligned}$$

where $k_0$ is the geodesic curvature under the metric $g_0$, and $\partial _{n_0}$ the outer-normal derivative with respect to to $g_0$.

Proof

We parameterize $\gamma$ by arclength in g and let N be the outer normal vector field on $\gamma$, namely $g ({\dot{\gamma }} , {\dot{\gamma }} ) = g (N, N) \equiv 1$. We have that ${\dot{\gamma }}_0 := e^{\sigma } {\dot{\gamma }}$ and $N_0 : = e^{\sigma } N$ are unit vectors under $g_0$. The geodesic curvature of $\partial D$ is given by

$$\begin{aligned} k_g = g \left( \nabla _{g,{\dot{\gamma }} }{\dot{\gamma }} , - N \right) . \end{aligned}$$

The covariant derivative $\nabla _g$ is related to the covariant derivative $\nabla _0$ under $g_0$ by

$$\begin{aligned} \nabla _{g, X} Y = \nabla _{0, X} Y + X(\sigma ) Y + Y(\sigma ) X - g_0 (X,Y) \nabla _0 \sigma . \end{aligned}$$

Therefore,

$$\begin{aligned} \nabla _{g,{\dot{\gamma }}}{\dot{\gamma }}&= \nabla _{0, {\dot{\gamma }} } {\dot{\gamma }} + 2g_0 ({\dot{\gamma }}, \nabla _0 \sigma ) {\dot{\gamma }} - g_0 ({\dot{\gamma }} ,{\dot{\gamma }}) \nabla _0 \sigma \\&= e^{-2\sigma } \nabla _{0, {\dot{\gamma }}_0 } {\dot{\gamma }}_0 + 2g_0 ({\dot{\gamma }}, \nabla _0 \sigma ) {\dot{\gamma }} - e^{-2\sigma } \nabla _0 \sigma . \end{aligned}$$

Since $g ({\dot{\gamma }}, N) = 0$, we have

$$\begin{aligned} g \left( \nabla _{g,{\dot{\gamma }} }{\dot{\gamma }}, - N \right)&= e^{2\sigma } g_0 \left( e^{-2\sigma } (\nabla _{0, {\dot{\gamma }}_0 } {\dot{\gamma }}_0 - \nabla _0 \sigma ), -e^{-\sigma } N_0\right) \\&= e^{-\sigma } (k_0 + \partial _{n_0} \sigma ) \end{aligned}$$

as claimed. $\square$