On Fuchs's additive intersection problem for the hyperbolic metric

TL;DR

This paper investigates the ratio of hyperbolic metrics on intersecting domains, proving the supremum is infinite in general but equals one for simply connected domains, and establishing the infimum as one-half with equality only when the domains are identical.

Contribution

It solves Fuchs's additive intersection problem for hyperbolic metrics, providing exact bounds and conditions for the ratio in various domain classes.

Findings

Supremum of the ratio is infinite for general hyperbolic domains.

Supremum of the ratio is 1 for simply connected domains.

Infimum of the ratio is 1/2, attained only when the domains are equal.

Abstract

For hyperbolic domains $D_1,D_2\subset \{z\in\mathbb C:|z|<R\}$ and $z\in D_1\cap D_2$, we consider the ratio $$ \frac{\lambda_{D_1\cap D_2}(z)} {\lambda_{D_1}(z)+\lambda_{D_2}(z)}. $$ We solve a problem of W. H. J. Fuchs by proving that the supremum of this ratio is $+\infty$ when $D_1$ and $D_2$ range over all hyperbolic domains. If $D_1$ and $D_2$ are further assumed to be simply connected, then the supremum is $1$. We also show that the infimum of this ratio is $\frac12$ in both settings, and that the value $\frac12$ is attained if and only if $D_1=D_2$.