Mapping-out function of a compact H-hull ($g_A$)

The mapping-out function is the unique conformal map $g_A:\mathbb{H}\setminus A\to \mathbb{H}$ with hydrodynamic normalization at infinity

For nested compact H-hulls whose closures avoid $x\in \mathbb{R}$, the boundary derivative is an excursion avoidance probability. Avoiding the larger hull is a subset of avoiding the smaller one, which proves the derivative inequality. For a hull contained in $\{|z|\le R\}$ and $|x|>R$, comparison with the filled half-disc gives the sharp bound $1-R^2/x^2\le g_A^{\prime }(x)\le 1$, since that half-disc has mapping-out function $z+R^2/z$. The empty hull and filled half-disc attain the two bounds.

If a compact H-hull lies in a disc of radius $r$ centred at $\xi \in \mathbb{R}$, the displayed estimate holds for $|z-\xi |>2r$, with an absolute constant $C$. It refines the Laurent expansion uniformly in the hull and supplies the first-order increment needed to derive the Chordal Loewner equation from the Loewner local growth property.

Let $D=\mathbb{H}\setminus K$ and stop planar Brownian motion from $z\in D$ on reaching height $R$ or leaving $D$. Then $\mathrm{Im}{g}_K(z)={\lim }_{R\to \infty }R{\mathbb{P}}_z(\mathrm{Im}{B}_{{\tau }_R}=R)$. The stopped harmonic height is a bounded martingale. The bounded difference $g_K(z)-z$ makes its top-boundary value differ from $R$ by a uniform constant, while optional stopping of the imaginary coordinate bounds the top exit probability by $\mathrm{Im}z/R$. Killing on the hull is essential.

For a compact H-hull, $v(z)=\mathrm{Im}{g}_K(z)$ is harmonic and satisfies $0<v(z)\le \mathrm{Im}z$. Local boundedness and the interior continuity of $g_K^{-1}$ show that $v$ tends to zero at every finite boundary point. The maximum principle for harmonic functions applied to $v-\mathrm{Im}z$, with an exhaustion controlling infinity through the Laurent expansion, proves the inequality without boundary smoothness.

The inverse of a hydrodynamically normalized mapping-out function has expansion $f(w)=w+O(1/w)$ at infinity. If bounded-domain points $z_n$ had unbounded images $w_n=g(z_n)$, then $z_n=f(w_n)=w_n+O(1/w_n)$ would be unbounded, a contradiction. Thus $g$ is bounded even near finite rough boundary points. Its own expansion at infinity then makes $g(z)-z$ uniformly bounded throughout the domain. Reflection is needed only near infinity, not along the entire hull boundary.

For the compact H-hull $(x,x+ih]$, the square-root branch asymptotic to $z-x$ gives the hydrodynamic normalization at infinity of its mapping-out function of a compact H-hull. Its expansion has coefficient $h^2/2$, the half-plane capacity of a vertical slit.

If a compact H-hull lies in the unit disc, compare its tail harmonic measure with that of the empty hull and filled half-disc. The Poisson kernel for the upper half-plane turns the inclusion of Brownian exit events into $x\le g_K(x)\le x+1/x$ for $x>1$. Reflection gives $x+1/x\le g_K(x)\le x$ for $x<-1$.

For a compact H-hull inside the disc of radius $r$ centred at a real point, its mapping-out function of a compact H-hull displaces every point of its domain by at most $3r$. After scaling and translation, the real boundary bounds for a unit-disc H-hull put the corresponding inverse boundary interval inside $[-2,2]$. Boundary cluster points of the inverse over this interval lie in the unit disc. The maximum modulus principle applied to $w-g_K^{-1}(w)$ gives the bound without assuming local connectedness of the hull. The nearly closed semicircular slit makes the constant three sharp.

With $m(z)=(z+1)/(1-z)$, the compact H-hull $K_L=m^{-1}(i(0,L])$ is a unit semicircular arc attached at $-1$, with a narrow passage near $1$. As $L\to \infty$, its normalized mapping-out function of a compact H-hull tends to $2$ at each fixed point of the interior bay. Points in that bay tending to $-1$ therefore have displacement approaching three. The order of these two limits is important.

A conformal map from an upper-half-plane domain has hydrodynamic normalization when $g(z)-z\to 0$ at infinity. For a compact H-hull this normalization makes its mapping-out function of a compact H-hull unique.

The half-plane capacity of a compact H-hull is the nonnegative coefficient $a$ in

The Brownian representation of half-plane capacity is $\mathrm{hcap}(K)={\lim }_{y\to \infty }y{\mathbb{E}}_{iy}\mathrm{Im}{B}_T$. The harmonic capacity from infinity in the upper half-plane uses $\mathrm{cap}(K)={\lim }_{y\to \infty }\pi y{\mathbb{P}}_{iy}(B_T\in K)$. Since $0\le \mathrm{Im}{B}_T\le \mathrm{rad}(K)1_{\{B_T\in K\}}$, the displayed bound follows. The normalization factor $\pi$ cannot be omitted. Independently, enclosure in a half-disc and monotonicity of half-plane capacity give the sharp bound $\mathrm{hcap}(K)\le \mathrm{rad}(K{)}^2$.

For $x\in \mathbb{R}$ and $R>0$, the filled half-disc $K=\{z\in \mathbb{H}:|z-x|\le R\}$ is a compact H-hull. Its mapping-out function of a compact H-hull is $g_K(z)=z+R^2/(z-x)$. On the semicircle the image is real, and the inverse branch asymptotic to the identity maps the complex upper half-plane onto the exterior half-disc. Its Laurent series has coefficient $R^2/z$, so $\boxed{\mathrm{hcap}(K)=R^2}$. Using an open half-disc instead requires taking its relative closure first.

Suppose a capacity-parameterized Loewner chain lies in a domain on which $\psi$ is a conformal map. For its image chain set ${\psi }_t={\tilde{g}}_t\circ \psi \circ g_t^{-1}$. Then the image Loewner driving function is ${\tilde{U}}_t={\psi }_t(U_t)$ andThe derivative is real by the Schwarz reflection principle. Cancellation of the pole in the differentiated conjugacy identity proves the squared-derivative rule.

For conjugated Loewner chains, ${\varphi }_t={\tilde{g}}_t\circ \varphi \circ g_t^{-1}$ and ${\tilde{\xi }}_t={\varphi }_t({\xi }_t)$. The image capacity speed is $2{\varphi }_t^{\prime }({\xi }_t{)}^2$. Differentiating this identity gives ${\dot{\varphi }}_t(z)=2{\varphi }_t^{\prime }({\xi }_t{)}^2/({\varphi }_t(z)-{\varphi }_t({\xi }_t))-2{\varphi }_t^{\prime }(z)/(z-{\xi }_t)$. The removable singularity at the driver has value $-3{\varphi }_t^{\prime \prime }({\xi }_t)$.

Let $h_t$ intertwine two Loewner chains, with driver ${\xi }_t=\sqrt{\kappa }{B}_t$, and set $a_t=h_t^{\prime }({\xi }_t)$, $b_t=h_t^{\prime \prime }({\xi }_t)$, $c_t=h_t^{\prime \prime \prime }({\xi }_t)$. Expanding the conjugacy equation at the driver gives ${\partial }_t{h}_t^{\prime }({\xi }_t)=b_t^2/(2a_t)-4c_t/3$; the Itô formula gives the displayed diffusion. For a real exponent $p$, the drift of $a_t^p$ is $p[1+\kappa (p-1)]a_t^{p-2}{b}_t^2/2+p(\kappa /2-4/3)a_t^{p-1}{c}_t$. At $\kappa =8/3$, $p=5/8$ cancels both terms and yields the SLE eight-thirds restriction martingale.

Apply the Itô formula to the transformed Loewner driving function ${\tilde{\xi }}_t={\varphi }_t(\sqrt{\kappa }{W}_t)$. The conformal conjugacy derivative for the chordal Loewner equation cancels the Itô drift exactly when $\kappa =6$. Its quadratic variation is $\kappa {\int }_0^t{\varphi }_s^{\prime }({\xi }_s{)}^{2}ds$.

For a vertical slit of height $h>0$ attached at $x\in \mathbb{R}$, the mapping-out function of a compact H-hull is $g(z)=x+\sqrt{(z-x{)}^2+h^2}$, with the branch asymptotic to $z-x$ at infinity. Its half-plane capacity is $h^2/2$.

For $\lambda >0$ and $x\in \mathbb{R}$,

If compact H-hulls satisfy $A\subseteq C$, thenThis follows from the Brownian representation of half-plane capacity or from composition of their mapping-out functions.

If compact H-hulls satisfy $A\subseteq C$ and $D$ is the bounded filling of $g_A(C\setminus A)$, thenIn particular, inclusion is strict exactly when the capacity increase is positive.

If $\tau$ is the first exit time of planar Brownian motion from $\mathbb{H}\setminus A$, thenIt follows by applying the optional sampling theorem for a supermartingale to the harmonic function $\mathrm{Im}(z-g_A(z))$.

There is a universal constant $C$ such that every compact H-hull satisfiesTranslate and scale so the hull lies in a unit half-disc, then use the Brownian representation of half-plane capacity and the $O(y^{-1})$ harmonic measure of that half-disc as viewed from $iy$.

For $R_r=[-r,r]\times (0,1]$ with $r\ge 1$,Indeed, the imaginary part at the Brownian exit point is at most one and the probability of reaching the radius-$O(r)$ neighbourhood containing the rectangle from $iy$ is $O(r/y)$. Consequently $r^{-1}{R}_r=[-1,1]\times (0,r^{-1}]$ has capacity $O(r^{-1})$ although its diameter tends to two.

A growing hull is parameterized by half-plane capacity when $\mathrm{hcap}(K_t)=2t$. The factor two makes its Chordal Loewner equation take the conventional form ${\partial }_t{g}_t=2/(g_t-U_t)$.

If $z_n\in \mathbb{H}\setminus A$ approaches a point of a compact H-hull $A$, thenThe real part need not converge: the two sides of a vertical slit have distinct real boundary values under its mapping-out function.