Surfaces with zero mean curvature H=0. First variation of area. Plateau problem. Classical examples: catenoid, helicoid. Costa surface (finite topology, embedded).
graph TD
D1["Def: Mean curvature H\nH = (κ₁+κ₂)/2"]
D2["Def: Minimal surface\nH ≡ 0"]
D3["Def: Weierstrass rep\nparametrization via holomorphic data"]
D4["Def: Finite topology\nfinite genus, ends"]
T1["Thm: First variation\nδArea = −2∫H N dA"]
T2["Thm: Minimal ⇔ conformal\nharmonic coord"]
T3["Thm: Bernstein (1912)\nplane only minimal graph in ℝ³"]
T4["Thm: Plateau problem\nexists disc spanning curve"]
T5["Ex: Catenoid, helicoid\ncomplete embedded"]
T6["Ex: Costa surface (1982)\nembedded, genus 1, 3 ends"]
D1 --> D2
D2 --> T1
D2 --> T2
T2 --> T3
T3 --> T4
T4 --> T5
T5 --> T6
D3 --> T5
classDef definition fill:#b197fc,color:#fff
classDef theorem fill:#51cf66,color:#fff
class D1,D2,D3,D4 definition
class T1,T2,T3,T4,T5,T6 theorem
Process Statistics
- Nodes: 16
- Edges: 14
Frontier: math.DG — minimal surfaces, embeddedness, moduli