Middelkurvaturen

Middelkurvaturen, in differential geometry often referred to as mean curvature, is a measure of how a surface bends on average at a point. For a smooth surface embedded in three-dimensional space, two principal curvatures k1 and k2 describe bending along orthogonal directions. The mean curvature H is defined as the average of these principal curvatures: H = (k1 + k2)/2. The sign of H depends on the choice of the surface normal, since reversing the orientation changes the sign.

Equivalently, H can be expressed as one-half the trace of the shape operator, or as one-half the divergence of the unit normal field: H = (1/2) div n. For a graph z = f(x, y) the mean curvature can be written as a second-order differential expression in f, reflecting how the surface bends in space.

Examples help illustrate the concept. A plane has H = 0 everywhere, a sphere of radius R has H = 1/R at every point, and a cylinder of radius R has H = 1/(2R). These values illustrate how curvature quantifies bending in different directions.

Middelkurvaturen is central in variational problems: surfaces that minimize area under a boundary constraint have zero mean curvature (they are minimal surfaces). Surfaces with constant mean curvature (CMC), such as spheres and cylinders, arise in physical contexts like soap films and droplets. The Young–Laplace equation relates the pressure difference across a surface to its mean curvature, reinforcing the physical relevance of H in interfaces and capillarity.

Extensions include mean curvature for higher-dimensional submanifolds and the mean curvature flow, where a surface evolves in the normal direction at a speed given by its mean curvature, used to smooth and straighten shapes.