On the Loewner Energy and Curve Composition
Time: 2025-10-09
Published By: He Liu
Speaker(s): Yaosong Yang (BICMR)
Time: 16:00-17:00 October 10, 2025
Venue: Room 77201, Jingchunyuan 78, BICMR
Loewner energy, arising as a good rate function for the large deviation principle of $\operatorname{SLE}$, bridges probability theory and Teichm\"uller theory. Heuristicly, this energy measures how round a Jordan curve is. The composition of Jordan curves in universal Teichm\"uller space $\gamma \circ \eta$ is defined through the composition of their conformal weldings $h_\gamma \circ h_\eta$ . We show that whenever $\gamma$ and $\eta$ are curves of finite Loewner energy $I^L$, the energy of the 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. This talk is based on a joint work with Tim Mesikepp.
