Conformal radius (${\mathrm{crad}}_D(z)=|{\varphi }^{\prime }(0)|$)

For a simply connected domain $D\ne \mathbb{C}$ and $z\in D$, the conformal radius is $|{\varphi }^{\prime }(0)|$, where $\varphi$ maps the unit disc conformally onto $D$ with $\varphi (0)=z$. Disc automorphisms fixing zero are rotations, so the value is independent of the chosen map.

Before the Loewner swallowing time of $z$, set $Z_t=g_t(z)-U_t=X_t+i{Y}_t$ and ${{\rm Y}}_t=Y_t/|g_t^{\prime }(z)|$. Since the conformal radius of the complex upper half-plane at $Z_t$ is $2Y_t$, the transformation rule gives $2{{\rm Y}}_t={\mathrm{crad}}_{D_t}(z)$. The Koebe quarter theorem and the interior-disc comparison give $\mathrm{dist}(z,\partial D_t)/2\le {{\rm Y}}_t\le 2\mathrm{dist}(z,\partial D_t)$. Differentiating the Chordal Loewner equation gives $\boxed{{\partial }_t\log {{\rm Y}}_t=-4Y_t^2/|Z_t{|}^4}$. Thus this quantity decreases as the remaining domain shrinks.