When minimal surfaces are stable, genus considerations, and the Costa surface
For over a century, mathematicians thought they had found every complete minimal surface without self-crossings and with finitely many holes and ends. Then in 1984, Celso Costa found one with a handle.
Click through the catenoid, helicoid, Enneper, and Scherk surfaces (genus 0, in cyan), then the Costa surface (genus 1, in violet), and compare their genus, ends, and year.
The first non-trivial minimal surface, discovered by Euler
Not all minimal surfaces are created equal. Some are stable — small perturbations increase area, so the surface sits in a true energy valley. Others are unstable — saddle points of the area functional that can lower their energy by deforming.
A minimal surface is stable if the second variation of area is non-negative for all compactly supported normal variations. The only complete stable minimal surface in R³ is the plane (do Carmo–Peng, Fischer-Colbrie–Schoen, 1979).
The topology of a minimal surface — its genus, number of ends, and embeddedness — profoundly constrains what shapes are possible. For over a century, mathematicians believed the plane, catenoid, and helicoid were the only complete, embedded minimal surfaces of finite topology. Then in 1984, Celso Costa shattered this belief.
The Costa surface (1984) is a complete, embedded minimal surface of genus 1 with three ends: two catenoidal and one planar. Its discovery opened the floodgates to the Costa–Hoffman–Meeks family of every genus.
| Genus | Ends | Examples |
|---|---|---|
| 0 | 1 | Plane, Helicoid |
| 0 | 2 | Catenoid |
| 0 | ∞ (periodic) | Scherk |
| 1 | 3 | Costa surface (1984) |
| k ≥ 1 | 3 | Costa–Hoffman–Meeks family |
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