On Fuchs's additive intersection problem for the hyperbolic metric

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 $D_1=D_2$.