Mean curvature

In mathematics, the mean curvature ${\displaystyle H}$ of a surface ${\displaystyle S}$ is an extrinsic measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space.

The concept was introduced by Sophie Germain in her work on elasticity theory. It is important in the analysis of minimal surfaces, which have mean curvature zero, and in the analysis of physical interfaces between fluids (such as soap films) which by the Young–Laplace equation have constant mean curvature.

Let ${\displaystyle p}$ be a point on the surface ${\displaystyle S}$. Each plane through ${\displaystyle p}$ containing the normal line to ${\displaystyle S}$ cuts ${\displaystyle S}$ in a (plane) curve. Fixing a choice of unit normal gives a signed curvature to that curve. As the plane is rotated by an angle ${\displaystyle \theta }$ (always containing the normal line) that curvature can vary. The maximal curvature ${\displaystyle \kappa _{1}}$ and minimal curvature ${\displaystyle \kappa _{2}}$ are known as the principal curvatures of ${\displaystyle S}$.

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