On Fuchs's additive intersection problem for the hyperbolic metric

Yixin He; Quanyu Tang

paper

On Fuchs's additive intersection problem for the hyperbolic metric

arXiv:2603.16676

Abstract

For hyperbolic domains $D_{1}, D_{2} \subset {z \in C : ∣ z ∣ < R}$ and $z \in D_{1} \cap D_{2}$ , we consider the ratio $\frac{I ^ » _{D_{1} \cap D_{2}} ( z )}{I ^ » _{D_{1}} ( z ) + I ^ » _{D_{2}} ( z )} .$ 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 $\frac{1}{2}$ in both settings, and that the value $\frac{1}{2}$ is attained if and only if $D_{1} = D_{2}$ .

14 pages. Comments and suggestions are welcome