---
tag: EF0QMQB2J1
kind: example
name: Costa's minimal surface
msc: [53A10]
instance_of: [41X0YW2R9N]
sources:
  - url: https://en.wikipedia.org/wiki/Costa's_minimal_surface
---

## Example

Costa's surface is a complete [[minimal surface|41X0YW2R9N]] in $\R^3$ that is embedded and has finite total curvature.
Topologically it is a torus with three points removed.
Near the three removed points the surface has three ends: two look like the ends of a [[catenoid|6NN1HP54GB]] and one looks like a plane.

It is built with the [[Weierstrass–Enneper parameterization|YWV6J23AP6]].
The domain is the square torus $\C / (\Z + i\Z)$ minus three points, and the data are given by the Weierstrass $\wp$ function of this lattice.
One checks that the real periods of the integrals vanish, so the surface is well defined, and minimality then follows from the parameterization.
The hard part, proved later, is that the surface has no self-intersections.

## Remarks

Costa described the surface in 1982, and Hoffman and Meeks proved that it is embedded.
Before it, the only known complete embedded minimal surfaces of finite topology were the plane, the catenoid and the helicoid.
Its total curvature is $-12\pi$.

## Explanations

### L1 human

Costa's surface looks like a catenoid with a flat sheet slicing through its waist, where the sheet and the tube are joined by two tunnels instead of crossing.
It has a hole like a doughnut and goes off to infinity in three directions.
Its discovery showed that minimal surfaces without self-crossings are much more varied than had been thought.