On Loewner energy and curve composition

Abstract

The composition $\gamma \circ \eta$ of Jordan curves $\gamma$ and $\eta$ in universal Teichm\"uller space is defined through the composition $h_\gamma \circ h_\eta$ of their conformal weldings. We show that whenever $\gamma$ and $\eta$ have finite Loewner energy $I^L$, the energy of their composition satisfies $$I^L(\gamma \circ \eta) \lesssim_K I^L(\gamma) + I^L(\eta),$$ with an explicit constant in terms of the quasiconformal $K$ of $\gamma$ and $\eta$. We also study the asymptotic growth rate of the Loewner energy under $n$ self-compositions $\gamma^n := \gamma \circ \cdots \circ \gamma$, showing $$\limsup_{n \rightarrow \infty} \frac{1}{n}\log I^L(\gamma^n) \lesssim_K 1,$$ again with explicit constant. Our approach is to define a new conformally-covariant rooted welding functional $W_h(y)$, and show $W_h(y) \asymp_K I^L(\gamma)$ when $h$ is a welding of $\gamma$ and $y$ is any root (a point in the domain of $h$). In the course of our arguments we also give several new expressions for the Loewner energy, including generalized formulas in terms of the Riemann maps $f$ and $g$ for $\gamma$ which hold irrespective of the placement of $\gamma$ on the Riemann sphere, the normalization of $f$ and $g$, and what disks $D, \overline{D}^c \subset \hatC$ serve as domains. An additional corollary is that $I^L(\gamma)$ is bounded above by a constant only depending on the Weil--Petersson distance from $\gamma$ to the circle.