Conformal = angle-preserving. Riemann mapping theorem (simply connected → disc). Uniformization. Riemann surfaces as one-dimensional complex manifolds.
graph TD
D1["Def: Conformal map\npreserves angles\nf'(z)≠0"]
D2["Def: Riemann surface\n1-dim C-manifold"]
D3["Def: Genus g\n topological invariant"]
T1["Thm: Riemann mapping\nsimply connected Ω≠ℂ\n⇒ Ω ≅ 𝔻"]
T2["Thm: Uniformization\nRiemann surf ⇒ ℂ̂,ℂ,𝔻"]
T3["Thm: Measurable RMT\nμ on 𝕊¹ ⇒ exists f:𝔻→Ω"]
T4["Thm: Koebe 1/4\nunivalent f, f(0)=0\n⇒ f(𝔻)⊃B(0,|f'(0)|/4)"]
T5["Thm: Carathéodory\nextension to boundary"]
D1 --> T1
D2 --> T2
D3 --> T2
T1 --> T3
T1 --> T4
T1 --> T5
classDef definition fill:#b197fc,color:#fff
classDef theorem fill:#51cf66,color:#fff
class D1,D2,D3 definition
class T1,T2,T3,T4,T5 theorem
Process Statistics
- Nodes: 13
- Edges: 11
- Definitions: 3
- Theorems: 5
Frontier: Teichmüller theory, extremal length, SLE, conformal field theory. math.CV