A minimal surface has vanishing mean curvature vector. For a smooth surface in three-dimensional Euclidean space, this means , equivalently . Thus Gaussian curvature is , but neither principal curvature needs to vanish. A plane and a catenoid both satisfy the condition, with very different Gaussian curvatures.
By the first variation of area formula, a minimal surface is a stationary point of surface area for every smooth variation that fixes its boundary, or has compact support in its interior. The word “minimal” specifies this differential equation; it does not guarantee a global minimum of surface area, or even a local minimum. A stable minimal surface imposes a further condition on the second variation.
A minimal surface is a surface that locally minimizes its area for a given boundary. More formally, a minimal surface is defined as a surface with a mean curvature of zero at every point. This means that, at each point on the surface, the surface is as flat as possible and does not bend upwards or downwards. Minimal surfaces can often be described using parametric equations or as graphs of functions.
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