---
tag: RPBKZ7E9VE
kind: concept
name: Constant-mean-curvature surface
msc: [53A10, 53C42]
specializes: [6080X3FAD4]
sources:
  - url: https://en.wikipedia.org/wiki/Constant-mean-curvature_surface
same_as:
  wikidata: Q5163611
  wikipedia: Constant-mean-curvature surface
---

## Summary

A constant-mean-curvature surface, or CMC surface, is a surface in space whose mean curvature takes the same value at every point.
Minimal surfaces are the case where this value is zero.
CMC surfaces are the critical points of area among surfaces that enclose a fixed volume.
So they model soap bubbles and other interfaces with a pressure difference.
Round spheres and circular cylinders are the first examples.

## Explanations

### L1 human

A soap bubble has the same pressure difference across its wall everywhere, so its mean curvature is the same everywhere.
That is why a free bubble is round.
A cylinder of radius $r$ is also a CMC surface: it bends with curvature $1/r$ around and not at all along, so the average is $1/(2r)$ at every point.