---
tag: PA78NFEPCF
kind: concept
name: Mean curvature
msc: [53A05, 53A10]
sources:
  - url: https://en.wikipedia.org/wiki/Mean_curvature
same_as:
  wikidata: Q1318998
  wikipedia: Mean curvature
---

## Summary

The mean curvature of a surface at a point is the average of its two [[principal curvatures|FA8QE1SGP4]].
Unlike the Gaussian curvature, it depends on how the surface sits in space, and its sign depends on the choice of unit normal.
Surfaces with mean curvature zero are the minimal surfaces, such as soap films, and surfaces of constant mean curvature model soap bubbles.

## Explanations

### L1 human

At a point of a surface, the bending can be different in different directions.
The mean curvature averages the strongest and the weakest bending.
A soap film spanning a wire loop has mean curvature zero: at each point it bends up in one direction exactly as much as it bends down in the other.
A soap bubble, a sphere of radius $r$, has the same mean curvature $1/r$ at every point, measured with the inner normal.