---
tag: 41X0YW2R9N
kind: concept
name: Minimal surface
msc: [53A10, 49Q05]
specializes: [RPBKZ7E9VE]
sources:
  - url: https://en.wikipedia.org/wiki/Minimal_surface
same_as:
  wikidata: Q1545397
  wikipedia: Minimal surface
---

## Summary

A minimal surface is a surface in space whose mean curvature is zero at every point.
Equivalently, small pieces of it are critical points of area: no small deformation that fixes the boundary changes the area to first order.
Soap films spanning a wire frame are physical models of minimal surfaces.
The subject links differential geometry with complex analysis, the calculus of variations and partial differential equations.

## Explanations

### L1 human

Dip a wire loop into soapy water and pull it out: the film that forms tries to have as little area as possible.
Its shape is a minimal surface.
At each point it bends upward in one direction exactly as much as it bends downward in the perpendicular direction, so the two bendings cancel.
A flat disc in a flat loop is the simplest example; two parallel rings give a waisted tube called a catenoid.