A submultiplicative property of the Carath\'eodory metric on planar domains

TL;DR

This paper establishes a submultiplicative inequality for the Carathéodory metric on planar domains, relating the metrics on individual domains, their intersection, and union, enhancing understanding of their geometric properties.

Contribution

It introduces a novel inequality connecting the Carathéodory metrics on intersecting and union domains in the plane, providing new insights into their geometric relationships.

Findings

Derived a submultiplicative inequality for the Carathéodory metric

Connected metrics on domains, intersections, and unions in the plane

Enhanced understanding of the geometric behavior of the Carathéodory metric

Abstract

Given a pair of smoothly bounded domains $D_1, D_2 \subset \mathbb C$, the purpose of this note is to obtain an inequality that relates the Carath\'{e}odory metrics on $D_1, D_2, D_1 \cap D_2$ and $D_1 \cup D_2$.

Equations

$$\pi :\mathbb{D}\to D$$

$${\lambda }_{\mathbb{D}}(z)|dz|=\frac{|dz|}{1-|z{|}^2}$$

$${\lambda }_D(\pi (z))|{\pi }^{\prime }(z)|={\lambda }_{\mathbb{D}}(z)$$

$${\lambda }_D(z)=|{\pi }^{\prime }(0){|}^{-1}.$$

$${\lambda }_{D_1\cap D_2}(z){\lambda }_{D_1\cup D_2}(z)\le {\lambda }_{D_1}(z){\lambda }_{D_2}(z)$$

$$\Delta \log {\lambda }_D(z)=4{\lambda }_D^2(z),$$

$$\Delta u-\mu (u)-f=0,$$

$$c_D(z)|dz|=\sup \{|f^{\prime }(z)|:f:D\to \mathbb{D}\text{holomorphic and}f(z)=0\}.$$

$$c_V(h(z))|h^{\prime }(z)|\le c_U(z),z\in U.$$

$$c_{D_1\cap D_2}(z)\cdot c_{D_1\cup D_2}(z)\le C{c}_{D_1}(z)\cdot c_{D_2}(z)$$

$${\kappa }_D(z)=-c_D^{-2}(z)\Delta \log c_D(z)$$

$$\underset{z\to p}{\lim }\frac{c_{U\cap D}(z)}{c_D(z)}=1.$$

$$-C\le {\kappa }_D(z)\le -4$$

$$c(z)|dz|=\frac{c_{D_1}(z)\cdot c_{D_2}(z)}{c_{D_1\cup D_2}(z)}|dz|$$

$$\begin{array}{rl}\kappa (z)& =-c^{-2}(z)\Delta \log c(z)\\ & =-c^{-2}(z)\left(\Delta \log c_{D_1}(z)+\Delta \log c_{D_2}(z)-\Delta \log c_{D_1\cup D_2}(z)\right)\\ & =I_1+I_2+I_{12},\end{array}$$

$$I_1=-c^{-2}(z)\Delta \log c_{D_1}(z),$$

$$I_2=-c^{-2}(z)\Delta \log c_{D_2}(z)$$

$$I_{12}=c^{-2}(z)\Delta \log c_{D_1\cup D_2}(z).$$

$$\kappa (z)\ge I_1+I_2.$$

$$I_1\ge {\kappa }_1(z)$$

$$I_2\ge {\kappa }_2(z).$$

$$\kappa (z)\ge {\kappa }_1(z)+{\kappa }_2(z)>-C$$

$$u(z)=\log\left(\frac{c_{\epsilon}(z)}{\sqrt{C\big{/}4}\,c(z)}\right)$$

$$\underset{z\to \xi }{\lim }\frac{c_{D_1}(z)\cdot c_{D_2}(z)}{c_{D_1\cup D_2}(z)}\ge \underset{z\to \xi }{\lim }{c}_{D_2}(z)=+\infty ,$$

$$\begin{array}{rl}0\ge \Delta u(z_0)& =\Delta \log c_{ϵ}(z_0)-\Delta \log c(z_0)\\ & \ge -{\kappa }_{ϵ}(z_0)c_{ϵ}^2(z_0)-C{c}^2(z_0)\\ & \ge 4c_{ϵ}^2(z_0)-C{c}^2(z_0),\end{array}$$

$$u(z_{0})=\log\left(\frac{c_{\epsilon}(z_{0})}{\sqrt{C\big{/}4}\,c(z_{0})}\right)\leq 0.$$

$$\log\left(\frac{c_{\epsilon}(z)}{\sqrt{C\big{/}4}\,c(z)}\right)\leq 0$$

$$c_{\epsilon}(z)\leq\sqrt{C\big{/}4}\,c(z)$$

$$c_{\epsilon}(z)\cdot c_{D_{1}\cup D_{2}}(z)\leq\sqrt{C\big{/}4}\,c_{D_{1}}(z)\cdot c_{D_{2}}(z)$$

$$c_{U}(z)\cdot c_{D_{1}\cup D_{2}}(z)\leq\sqrt{C\big{/}4}\,c_{D_{1}}(z)\cdot c_{D_{2}}(z)$$

Full text

1. Introduction

Let $\lambda_{D}(z)|dz|$ denote the Poincaré metric on a hyperbolic domain $D\subset\mathbb{C}$. To quickly recall the construction of this metric, note that there exists a holomorphic covering from the unit disc $\mathbb{D}$ to $D$,

[TABLE]

whose deck transformations form a Fuchsian group $G$ that acts on $\mathbb{D}$. The Poincaré metric on $\mathbb{D}$,

[TABLE]

is invariant under ${\rm Aut}(\mathbb{D})$ (hence in particular $G$) and therefore, for $z\in D$, the prescription

[TABLE]

defines the Poincaré metric $\lambda_{D}$ on $D$ in an unambiguous manner. For an arbitrary $z\in D$, we may choose the covering projection so that $\pi(0)=z$ and hence (1.1) implies that

[TABLE]

Solynin [SolyninMultiplicativity], [Solynin2] proved the following remarkable relation between the Poincaré metrics on a pair of hyperbolic domains and those on their union and intersection:

Let $D_{1},D_{2}\subset\mathbb{C}$ be domains such that $D_{1}\cap D_{2}\neq\emptyset$. Suppose that $D_{1}\cup D_{2}$ is hyperbolic. Then

[TABLE]

for all $z\in D_{1}\cap D_{2}$. If equality holds at one point $z_{0}\in D_{1}\cap D_{2}$, then $D_{1}\subset D_{2}$ or $D_{2}\subset D_{1}$ and in this case, equality holds for all points $z\in D_{1}\cap D_{2}$.

A direct proof of this was given by Kraus-Roth [OliverRothMultiplicativity] that relied on a computation reminiscent of the classical Ahlfors lemma and the fact that

[TABLE]

which precisely means $\lambda_{D}(z)|dz|$ has constant curvature $-4$ on $D$. Solynin’s result follows from a comparison result for solutions to non-linear elliptic PDE’s of the form

[TABLE]

where $\mu:\mathbb{R}\rightarrow\mathbb{R}$ and $f:D\rightarrow\mathbb{R}$ are suitable continuous non-negative functions that satisfy an additional convexity condition. Clearly, this reduces to the curvature equation (1.3) by writing $u=\log\lambda_{D}$ and letting $\mu(x)=4e^{2x}$ and $f\equiv 0$.

The purpose of this note is to prove an analogue of Solynin’s theorem for the Carathéodory metric on planar domains. Let $D\subset\mathbb{C}$ be a domain that admits at least one non-constant bounded holomorphic function. Recall that for $z\in D$, the Carathéodory metric $c_{D}(z)|dz|$ is defined by

[TABLE]

This is a distance-decreasing (and hence conformal) metric in the sense that if $h:U\rightarrow V$ is a holomorphic map between a pair of planar domains $U,V$, then

[TABLE]

If $U\subset V$, applying this to the inclusion $i:U\rightarrow V$ shows that the Carathéodory metric is monotonic as a function of the domain, i.e., $c_{V}(z)\leq c_{U}(z)$ for $z\in U$. Furthermore, for each $\zeta\in D$, there is a unique holomorphic map $f_{\zeta}:D\rightarrow\mathbb{D}$ that realizes the supremum in the definition of $c_{D}(z)$ – this is the Ahlfors map.

2. Statement and proof of the main result

Theorem 2.1**.**

Let $D_{1},D_{2}\subset\mathbb{C}$ be smoothly bounded domains with $D_{1}\cap D_{2}\neq\emptyset$. Then there exists a constant $C=C(D_{1},D_{2})>0$ such that

[TABLE]

for all $z\in D_{1}\cap D_{2}$.

Note that $D_{1}\cap D_{2}$ may possibly have several components. The notation $c_{D_{1}\cap D_{2}}(z)$ refers to the Carathéodory metric on that component of $D_{1}\cap D_{2}$ which contains a given $z\in D_{1}\cap D_{2}$. Furthermore, the constant $C>0$ is independent of which component of $D_{1}\cap D_{2}$ is being considered and only depends on $D_{1}$ and $D_{2}$. The proof, which is inspired by [OliverRothMultiplicativity], uses the following known supplementary properties of the Carathéodory metric:

First, $c_{D}(z)$ is continuous and $\log c_{D}(z)$ is subharmonic on $D$ – see [AhlforsAndBeurling] for instance. The possibility that $\log c_{D}(z)\equiv-\infty$ can be ruled out for bounded domain since for every $\zeta\in D$, the affine map $f(z)=(z-\zeta)/a$ vanishes at $\zeta$ and maps $D$ into the unit disc $\mathbb{D}$ for every positive $a$ that is bigger than the diameter of $D$. The fact that $f^{\prime}(z)\equiv 1/a>0$ shows that $c_{D}(\zeta)>0$ and hence $\log c_{D}(\zeta)>-\infty$.

Second, Suita [SuitaI] showed that $c_{D}(z)$ is real analytic in fact and hence we may speak of its curvature

[TABLE]

in the usual sense. The subharmonicity of $\log c_{D}(z)$ already implies that $\kappa_{D}\leq 0$ everywhere on $D$, but by using the method of supporting metrics, Suita [SuitaI] was also able to prove a much stronger inequality namely, $\kappa_{D}\leq-4$ on $D$. Following this line of inquiry further, Suita [SuitaII] (see [BurbeaPaper] as well) showed that if the boundary of $D$ consists of finitely many Jordan curves, the assumption that $\kappa_{D}(\zeta)=-4$ for some point $\zeta\in D$, implies that $D$ is conformally equivalent to $\mathbb{D}$.

Finally, it is known that this metric admits a localization near $C^{\infty}$-smooth boundary points – see for example [InvariantMetricJarnicki] which contains a proof for the case of strongly pseudoconvex points that works verbatim in the planar case too. That is, if $p\in\partial D$ is a $C^{\infty}$-smooth boundary point of a bounded domain $D\subset C$, then for a small enough neighbourhood $U$ of $p$ in $\mathbb{C}$,

[TABLE]

Using this, it was shown in [SV] that the curvature $\kappa_{D}(z)$ of the Carathéodory metric approaches $-4$ near each $C^{\infty}$-smooth boundary point of a bounded domain $D\subset\mathbb{C}$. The point here being that as one moves nearer to such a point, the metric begins to look more and more like the Carathéodory metric on $\mathbb{D}$ – the use of the scaling principle makes all this precise. It follows on any smoothly bounded planar domain $D$, $\kappa_{D}\approx-4$ for points close to the boundary and for those that are at a fixed positive distance away from it, there is a large negative lower bound for $\kappa_{D}$ as a result of its continuity. Hence for every such $D\subset\mathbb{C}$, there is a constant $C=C(D)>0$ such that

[TABLE]

for all $z\in D$. Another consequence of the localization principle is that $c_{D}(z)\rightarrow+\infty$ as $z$ approaches a smooth boundary point. Indeed, near such a point, $c_{D}(z)\approx c_{U\cap D}(z)$ and the latter is the same as the Carathéodory metric on the disc $\mathbb{D}$ (since $U\cap D$ can be taken to be simply connected) which blows up near every point on $\partial\mathbb{D}$.

To prove the theorem, let $\kappa_{1}$ and $\kappa_{2}$ be the curvatures of the Carathéodory metric $c_{D_{1}}(z)\lvert dz\rvert$ and $c_{D_{2}}(z)\lvert dz\rvert$ on $D_{1}$ and $D_{2}$ respectively.

Consider the metric

[TABLE]

on the possibly disconnected open set $D_{1}\cap D_{2}$. What follows applies to each component of $D_{1}\cap D_{2}$ without any regard to its analytic or topological properties and hence we will continue to write $c(z)\lvert dz\rvert$ to denote this metric on any given component. Its curvature is

[TABLE]

where

[TABLE]

[TABLE]

and

[TABLE]

Note that $I_{12}\geq 0$ since $\log c_{D_{1}\cup D_{2}}(z)$ is subharmonic and hence

[TABLE]

To analyze each of these terms, note that $c_{D_{1}}\geq c_{D_{1}\cup D_{2}}$ and $c_{D_{2}}\geq c_{D_{1}\cup D_{2}}$. Combining this with the fact that the curvature of the Carathéodory metric is negative everywhere, it follows that

[TABLE]

and

[TABLE]

Hence there is a constant $C=C(D_{1},D_{2})>0$ such that

[TABLE]

for all $z\in D_{1}\cap D_{2}$.

Let $U$ be a component of $D_{1}\cap D_{2}$. Since $D_{1},D_{2}$ have smooth boundaries, $U$ has finite connectivity, say $m\geq 1$ and is non-degenerate in the sense that the interior of its closure coincides with itself. In particular, its boundary cannot contain isolated points. Let $U_{\epsilon}$ be an $\epsilon$-thickening of $U$. For all sufficiently small $\epsilon>0$, $U_{\epsilon}$ also has connectivity $m$ and is non-degenerate. Furthermore, $U_{\epsilon}\rightarrow U$ in the sense of Carathéodory as $\epsilon\rightarrow 0$.

For a fixed $\epsilon>0$, let $c_{\epsilon}(z)|dz|$ be the Carathéodory metric on $U_{\epsilon}$. Consider the function

[TABLE]

on $U$. Since $U$ is compactly contained in $U_{\epsilon}$, $c_{\epsilon}$ is bounded on $U$ and if $\xi\in\partial U$, then as $z\rightarrow\xi$ within $U$,

[TABLE]

where the inequality follows from the monotonocity of the Carathéodory metric, i.e., $c_{D_{1}}(z)\geq c_{D_{1}\cup D_{2}}(z)$. The fact that $\xi$ is a smooth boundary point of $D_{2}$ implies that $c_{D_{2}}$ blows up near it and this means that $u(z)\to-\infty$ at $\partial U$. Therefore, $u$ attains a maximum at some point $z_{0}\in U$. As a result,

[TABLE]

where $\kappa_{\epsilon}$ is the curvature of $c_{\epsilon}(z)|dz|$. It follows that

[TABLE]

For an arbitrary $z\in U$, $u(z)\leq u(z_{0})\leq 0$ and this gives

[TABLE]

or

[TABLE]

which is same as

[TABLE]

and this holds for all $z\in U$. It remains to show that $c_{\epsilon}\rightarrow c_{U}$ pointwise on $U$ for then we can pass to the limit as $\epsilon\rightarrow 0$, keeping in mind that $C$ is independent of $\epsilon$, to get

[TABLE]

as claimed. Fix $p\in U$. To show that $c_{\epsilon}(p)\rightarrow c_{U}(p)$, it suffices to prove that $|f^{\prime}_{\epsilon}(p)|\rightarrow|f^{\prime}(p)|$ as $\epsilon\rightarrow 0$, where $f_{\epsilon}:U_{\epsilon}\rightarrow\mathbb{D}$ and $f:U\rightarrow\mathbb{D}$ are the respective Ahlfors maps for the domains $U_{\epsilon}$ and $U$ at $p$. What this is really asking for is the continuous dependence of the Ahflors maps on the domains. But this is addressed in [Younsi] – indeed, Theorem 3.2 therein can be applied as the domains $U_{\epsilon},U$ have the same connectivity by construction and the $U_{\epsilon}$’s decrease to $U$ as $\epsilon\rightarrow 0$. The nuance, in this theorem, about the base point being the point at infinity can be arranged by sending $p\mapsto\infty$ by the map $T(z)=1/(z-p)$ and working with the domains $T(U_{\epsilon})$ and $T(U)$. This completes the proof.

Question: Let $\Omega\subset\mathbb{C}$ be a domain on which the Carathéodory metric $c_{\Omega}(z)|dz|$ is not degenerate. Does its curvature $\kappa_{\Omega}$ admit a global lower bound?