The best bound of the area--length ratio in Ahlfors Covering surface theory (I)
| dc.creator | Zhang, Guang Yuan | |
| dc.date | 2009-03-20 | |
| dc.date | 2009-03-24 | |
| dc.date.accessioned | 2026-07-07T12:55:29Z | |
| dc.date.available | 2026-07-07T12:55:29Z | |
| dc.description | In Ahlfors' covering surface theory, it is well known that there exists a positive constant $h$ such that for any nonconstant holomorphic mapping $f:% \barΔ\to S,$ if $f(Δ)\cap \{0,1,\infty \}=\emptyset ,$ then% A(f,Δ)\leq hL(f,\partial Δ),% where $Δ$ is the disk $|z|<1$ in $\mathbb{C},$ $S$ is the unit Riemann sphere, $A(f,Δ)$ is the area of the image of $Δ$ and $% L(f,\partial Δ)$ is the length of the image of $\partial Δ$, both counting multiplicities. In this paper, we will show that the best lower bound for $h$ is the number h_{0}=\max_{τ\in \lbrack 0,1]}[ \frac{\sqrt{1+τ^{2}}(π+\arcsin τ)}{\mathrm{{arccot}\frac{\sqrt{1-τ^{2}}}{\sqrt{% 1+τ^{2}}}}}-τ] =4. \allowbreak 034 159 790 \allowbreak 51..., % and this is the exact estimation, i.e. there exists a sequence of holomorphic mappings $f_{n}:\barΔ\to S$ such that $% f_{n}(Δ)\cap \{0,1,\infty \}=\emptyset $ and \lim_{n\to \infty}A(f_{n},Δ)/L(f_{n},\partial Δ)=h_{0}. | |
| dc.description | 83 pages | |
| dc.identifier | https://arxiv.org/abs/0903.3460 | |
| dc.identifier | http://arxiv.org/abs/0903.3460 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/224266 | |
| dc.subject | Complex Variables | |
| dc.subject | Differential Geometry | |
| dc.subject | 30D35, 30D45, 52B60 | |
| dc.title | The best bound of the area--length ratio in Ahlfors Covering surface theory (I) | |
| dc.type | text |