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SUMMARY:Guang-Yuan Zhang (Tsinghua University)
DTSTART:20231128T140000Z
DTEND:20231128T150000Z
DTSTAMP:20260422T201823Z
UID:CAvid/114
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/CAvid/114/">
 The precise form of Ahlfors' Second Fundamental Theorem of covering surfac
 es</a>\nby Guang-Yuan Zhang (Tsinghua University) as part of CAvid: Comple
 x Analysis video seminar\n\nLecture held in N/A.\n\nAbstract\nA simply con
 nected covering surface $\\Sigma =\\left( f\,\\overline{\\Delta }%\n\\righ
 t) $ over the unit Riemann sphere $S$ is an orientation-preserving\,\ncont
 inuous\, open and finite-to-one mapping (OPCOFOM) $f$ from the closed\nuni
 t disk $\\overline{\\Delta }$ into the sphere $S$. Here open means that $f
 $\ncan be extended continuous and open to a neighborhood of $\\overline{\\
 Delta }.\n$ We denote by $\\mathbf{F}$ all simply connected surfaces.\n\nL
 et $E_{q}=\\left\\{ a_{1}\,a_{2}\,\\dots \,a_{q}\\right\\} $ be a set on t
 he unit\nRiemann sphere consisting of $q$ distinct points with $q>2.$ \nAh
 lfors' second\nfundamental theorem (SFT) states that there exists a positi
 ve number $h$\ndepending only on $E_{q}\,$ such that for any surface $\\Si
 gma =\\left( f\,%\n\\overline{\\Delta }\\right) \\in \\mathbf{F}\,$\n\\[\n
 \\left( q-2\\right) A\\left( \\Sigma \\right) <4\\pi \\overline{n}\\left( 
 \\Sigma\n\\right) +hL\\left( \\partial \\Sigma \\right) \,\n\\]\nwhere $\\
 Delta $ is the unit disk\, $A\\left( \\Sigma \\right) $ is the spherical\n
 area of $\\Sigma $\, $L\\left( \\partial \\Sigma \\right) $ is the spheric
 al\nlength of the boundary curve $\\partial \\Sigma =\\left( f\,\\partial 
 \\Delta\n\\right) \,$ and $\\overline{n}\\left( \\Sigma \\right) =\\#f^{-1
 }(E_{q})\\cap\n\\Delta .$\n\nIf we define $R\\left( \\Sigma \\right) =R\\l
 eft( \\Sigma \,E_{q}\\right) $ to be\nthe error term in Ahlfors' SFT\, say
 \,\n\\[\nR\\left( \\Sigma \\right) =\\left( q-2\\right) A\\left( \\Sigma \
 \right) -4\\pi\n\\overline{n}\\left( \\Sigma \\right) \,\n\\]\nthen Ahlfor
 s' SFT reads\n\\[\nH_{0}=\\sup_{\\Sigma \\in \\mathbf{F}}\\left\\{ \\frac{
 R(\\Sigma )}{L(\\partial\n\\Delta )}:\\Sigma =\\left( f\,\\overline{\\Delt
 a }\\right) \\right\\} <+\\infty .\n\\]\nWe call $H_{0}=H_{0}(E_{q})$ Ahlf
 ors' constant for simply connected\nsurfaces.\n\nIn this talk\, I will int
 roduce my recent work which identify the precise\nbound $H_{0}=H_{0}(E_{q}
 ).$\n
LOCATION:https://researchseminars.org/talk/CAvid/114/
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