---
tag: T2HPG1780R
kind: definition
name: Mean curvature
msc: [53A05, 53A10]
defines: PA78NFEPCF
uses: [6080X3FAD4, FA8QE1SGP4, DQCD11X1VM, MAJ3DH7MZS, 9KHQR16P1B, 07PTPSZPME]
sources:
  - url: https://en.wikipedia.org/wiki/Mean_curvature
---

## Definition

Let $S \subseteq \R^3$ be a [[regular surface|6080X3FAD4]], let $p \in S$, and choose a smooth unit normal field $N$ near $p$.
Let $S_p$ be the [[shape operator|DQCD11X1VM]] at $p$ and let $k_1, k_2$ be the [[principal curvatures|FA8QE1SGP4]] at $p$.
The *mean curvature* of $S$ at $p$ is
$$H(p) = \frac{k_1 + k_2}{2} = \frac{1}{2} \tr S_p.$$

## Remarks

In a local parametrisation, with $E, F, G$ the coefficients of the [[first fundamental form|MAJ3DH7MZS]] and $e, f, g$ those of the [[second fundamental form|9KHQR16P1B]],
$$H = \frac{e G - 2 f F + g E}{2 (E G - F^2)}.$$
Example: the cylinder of radius $r$ with the inner normal has $k_1 = 1/r$ and $k_2 = 0$, so $H = 1/(2r)$.
The saddle $z = xy$ has $k_1 = 1$ and $k_2 = -1$ at the origin, so $H = 0$ there.
Non-example: replacing $N$ by $-N$ changes $H$ into $-H$, so on a surface without a global unit normal, such as a Möbius strip, only $\abs{H}$ is defined globally.
Some authors define $H$ as $k_1 + k_2$, without the factor $\frac{1}{2}$.