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The extremal family of disc-valued univalent maps fixing a basepoint
Definition
Let be a homologically simply connected complex domain and let . The Riemann extremal family at is
The derivative condition fixes the rotational ambiguity after the basepoint normalization.
Depends on
Used by
- The unit-disc extremal problem is solved by the identity Example
- A maximizing sequence has a locally uniform limit with extremal derivative Lemma
- A proper homologically simply connected plane domain has a bounded univalent competitor Lemma
- The extremal derivatives are positive and have a finite supremum Lemma
Dependency tree · two levels
3 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Matthias Weber, Complex Analysis, §5.2 (standard reference, not scraped)
- Walter Rudin, Real and Complex Analysis, Theorem 14.9 (standard reference, not scraped)