---
tag: R7WFJGJBV4
kind: concept
name: Willmore energy
msc: [53A10, 53C42]
sources:
  - url: https://en.wikipedia.org/wiki/Willmore_energy
same_as:
  wikidata: Q2581692
  wikipedia: Willmore energy
---

## Summary

The Willmore energy of a closed surface in space is the integral of the square of its mean curvature over the surface.
It measures how far the surface is from being round: it is at least $4\pi$, with equality only for round spheres.
It does not change under scaling, and more generally under Möbius transformations of space.
It appears in the study of elastic membranes, such as cell membranes, where bending costs energy.

## Explanations

### L1 human

Think of a closed surface as a thin elastic shell; bending it costs energy.
The Willmore energy adds up, over the whole surface, the square of how much it bends on average at each point.
A round ball is the most relaxed shape, with energy $4\pi$ whatever its size.
A long thin sausage or a wrinkled surface has much larger energy.