On Fuchs's additive intersection problem for the hyperbolic metric
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912971121229824 |
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| author | He, Yixin Tang, Quanyu |
| author_facet | He, Yixin Tang, Quanyu |
| contents | 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 $D_1=D_2$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_16676 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On Fuchs's additive intersection problem for the hyperbolic metric He, Yixin Tang, Quanyu Complex Variables Primary 30F45, Secondary 30C20, 31A15 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 $D_1=D_2$. |
| title | On Fuchs's additive intersection problem for the hyperbolic metric |
| topic | Complex Variables Primary 30F45, Secondary 30C20, 31A15 |
| url | https://arxiv.org/abs/2603.16676 |