Koebe uniformization of nondegenerate domains with bounded gap-ratios

Key Points

• The central aim is to establish the conformal equivalence of nondegenerate domains with bounded gap-ratios to circle domains using extremal length.
• Utilized transboundary extremal length as a key analytical tool
• Focused on nondegenerate, uncountably connected domains
• Demonstrated properties of bounded gap-ratios
• Every nondegenerate domain with a bounded gap-ratio is shown to be conformally equivalent to a circle domain
• Established a clear relationship in complex analysis through this uniformization technique