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On Fuchs's additive intersection problem for the hyperbolic metric

Yixin He, Quanyu Tang

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 $$ \frac{λ_{D_1\cap D_2}(z)} {λ_{D_1}(z)+λ_{D_2}(z)}. $$ 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 $D_1=D_2$.

On Fuchs's additive intersection problem for the hyperbolic metric

Abstract

For hyperbolic domains and , we consider the ratio We solve a problem of W. H. J. Fuchs by proving that the supremum of this ratio is when and range over all hyperbolic domains. If and are further assumed to be simply connected, then the supremum is . We also show that the infimum of this ratio is in both settings, and that the value is attained if and only if .
Paper Structure (11 sections, 16 theorems, 135 equations)

This paper contains 11 sections, 16 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Related multiplicative intersection inequalities
  3. Paper organization
  4. Preliminaries
  5. Basic facts about the hyperbolic metric
  6. Intersection components of simply connected domains
  7. Logarithmic capacity and inversion
  8. The general upper bound is infinite
  9. The sharp upper bound in the simply connected case
  10. Sharp lower bounds and equality cases
  11. Concluding remarks and open problems

Key Result

Theorem 1.2

For every $R>0$, one has

Theorems & Definitions (35)

  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 25 more