2009/03/20 by Guang Yuan Zhang, Zhang, Guang Yuan
Computer Science · Mathematics · #30D35 #30D45 #52B60 #Advanced Graph Theory Research #Complex Variables (math.CV) #Computational Geometry and Mesh Generation #Differential Geometry (math.DG) #FOS: Mathematics #math.CV #math.DG #msc:30D35 #msc:30D45 #msc:52B60
paper · pdf · doi:10.48550/arxiv.0903.3460
83 pages
openalex publication_date 2009/03/20 · arxiv created 2009/03/24 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In Ahlfors' covering surface theory, it is well known that there exists a positive constant h such that for any nonconstant holomorphic mapping f:% Δ→ S, if f(Δ)∩ \0,1,∞ \=∅ , then% A(f,Δ)≤ hL(f,∂ Δ),% where Δ is the disk |z|<1 in ℂ, S is the unit Riemann sphere, A(f,Δ) is the area of the image of Δ and % L(f,∂ Δ) is the length of the image of ∂ Δ, both counting multiplicities. In this paper, we will show that the best lower bound for h is the number h0=maxτ∈ \lbrack 0,1][ \frac√1+τ2(π+\arcsin τ)\mathrmarccot\frac√1-τ2√% 1+τ2-τ] =4. \allowbreak 034 159 790 \allowbreak 51..., % and this is the exact estimation, i.e. there exists a sequence of holomorphic mappings fn:Δ→ S such that % fn(Δ)∩ \0,1,∞ \=∅ and limn→ ∞A(fn,Δ)/L(fn,∂ Δ)=h0.