Past exam of the mathematics course of the University of Cambridge / 2017 / iii / Paper 203 / 2 / a

A family $(K_t{)}_{t\ge 0}$ of compact H-hulls is non-decreasing whenThis permits equal hulls at different times. A Loewner chain with strictly increasing half-plane capacity has strict inclusion for $s<t$. Usually its initial compact H-hull is $K_0=\varnothing$.

Write $g_s=g_{K_s}$ and let $K_{s,t}$ be the relatively closed hull obtained by mapping the increment $K_t\setminus K_s$ through $g_s$. For a bounded compact H-hull $L$, defineThe Loewner local growth property isOne can equivalently use diameter: for a nonempty compact H-hull, $\mathrm{diam}(L)\le 2\mathrm{rad}(L)\le 2\mathrm{diam}(L)$, using a point of its closure on $\mathbb{R}$ for the second inequality. The condition concerns the mapped increment, not the original difference $K_{t+h}\setminus K_t$, which can be large when a narrow opening closes. It makes growth occur near one continuous Loewner driving function rather than by a sudden addition at separated boundary points.

With the normalization used in the Chordal Loewner equation, half-plane-capacity parameterization meansThus $K_0$ is empty and the half-plane-capacity composition rule gives $\mathrm{hcap}(K_{s,t})=2(t-s)$. The factor two produces ${\partial }_t{g}_t(z)=2/(g_t(z)-U_t)$. A convention using half-plane capacity $t$ instead changes that coefficient to one and rescales time; the convention must be kept consistent.