Minimal surface

A surface stationary for area under small admissible deformations. In a smooth interior region, its mean curvature is zero.

Mean curvature: The average of a surface’s principal curvatures at a point. Its vanishing characterizes smooth minimal surfaces away from boundaries.
Surface: A two-dimensional manifold embedded in or immersed in a larger space. Minimal surfaces are a geometrically constrained class of surfaces.
Catenoid: A surface of revolution formed by rotating a catenary around its directrix. It is the simplest nonplanar minimal surface of revolution.
Soap film: A thin liquid film bounded by a wire or other boundary. Surface tension drives a soap film toward a minimal-area configuration.
Least-area surface: A surface attaining the smallest area among surfaces satisfying specified constraints. Every smooth least-area surface is stationary, but a stationary surface need not be globally least-area.
First variation of area: The first-order change in a surface’s area under a deformation. Minimality means this change vanishes for every admissible variation.
Area: A measure of the two-dimensional size of a surface. It is the quantity minimal surfaces make stationary.
Helicoid: A ruled surface swept out by a line rotating while moving along an axis. It is a nonplanar minimal surface with screw symmetry.
Surface tension: The energy per unit area associated with a fluid interface. For uniform tension, minimizing interfacial energy reduces to minimizing area.
Constant mean curvature surface: A surface whose mean curvature has the same value at every point. It generalizes minimal surfaces, which have constant mean curvature equal to zero.

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