Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 203 / 1

A compact H-hull is a bounded, relatively closed set $A\subset \mathbb{H}$ for which $\mathbb{H}\setminus A$ is simply connected domain. Its mapping-out function of a compact H-hull is the unique conformal map $g_A:\mathbb{H}\setminus A\to \mathbb{H}$ with hydrodynamic normalization at infinityThe half-plane capacity is

Let $g=g_A$ and define $u(z)=\mathrm{Im}(z-g(z))$. This is harmonic on $\mathbb{H}\setminus A$, has boundary values $\mathrm{Im}z$ on the hull boundary and zero on the real boundary, and tends to zero at infinity. The representation by harmonic measure and optional sampling theorem therefore giveAt $z=iy$, the hydrodynamic expansion givesConsequently the Brownian representation of half-plane capacity is

Map out $A$ first. The imagewith its bounded filling is a compact H-hull, and uniqueness of hydrodynamic normalization givesComparing the coefficients of $1/z$ at infinity yields the half-plane-capacity composition ruleThus half-plane capacity is monotone under inclusion.

The statement is true. In the notation of part (ii), equality of the capacities forces $\mathrm{hcap}(D)=0$. Every nonempty compact H-hull has strictly positive half-plane capacity: by the Brownian representation of half-plane capacity, Brownian motion started sufficiently high has positive harmonic measure of a boundary portion of positive height. Hence $D=\varnothing$, so $C\setminus A=\varnothing$ and

The family is a nondecreasing family of sets when

It has the half-plane-capacity parameterization under the standard chordal convention when

It has the Loewner local growth property when, after mapping out the old hull, each short new increment is small: for every $T,ϵ>0$ there is $\delta >0$ such thatEquivalent formulations use a crosscut of diameter below $ϵ$ separating the new increment from infinity.

Write $r_t=\sqrt{2t}$. The hulls $A_t=r_t(\mathbb{H}\cap \mathbb{D})$ are nested, so property (i) holds. Scaling the given map givesso $\mathrm{hcap}(A_t)=2t$ and property (ii) holds.

Property (iii) fails. For $s>0$, the image under $g_s$ of the outer semicircle of $A_t\setminus A_s$ isAs $t\downarrow s$, this converges to $2r_s\cos \theta$, which fills the real interval $[-2r_s,2r_s]$. Hence the diameter of the mapped new increment tends to $4r_s$, rather than zero. Therefore (i) and (ii) hold, while (iii) does not.