a minimal surface
Dip a bent wire loop into soapy water and pull it out: the film that spans the loop pulls itself taut into the smallest possible area it can have with that boundary. The graceful, often saddle-shaped sheets you get are minimal surfaces. They are nature's way of answering 'what is the least-area surface with this rim?', and they appear wherever a surface wants to shrink itself under uniform tension.
Precisely, a minimal surface is one whose mean curvature vanishes at every point: H = (k_1 + k_2)/2 = 0, equivalently k_1 = -k_2 everywhere. So at each point the surface curves up exactly as much in one principal direction as it curves down in the perpendicular one — every point is a balanced saddle (or flat). This is the condition that makes the surface a critical point of area: if you perturb a minimal surface while holding its boundary fixed, the area does not change to first order. (That is why H = 0 is the natural name 'minimal,' though strictly it means stationary area, not always least — see the caution.) Since k_1 = -k_2, the Gaussian curvature K = k_1*k_2 = -k_1^2 is always less than or equal to zero, so minimal surfaces are saddle-shaped or flat at every point, never dome-like.
Minimal surfaces sit at the crossroads of geometry, analysis, and physics. They model soap films and tense membranes, they motivated the classical 'Plateau problem' (does a least-area surface always span a given closed curve? — yes, proved by Douglas and Rado), and modern examples like the helicoid, catenoid, and Costa's surface are landmarks of the field. Two honest cautions. First, H = 0 means stationary area, not guaranteed minimum — a piece of minimal surface only truly minimises area if it is small enough; large pieces can be unstable, like a soap film that suddenly snaps to a different shape. Second, a minimal surface is NOT flat: the plane is the trivial example, but the interesting ones (catenoid, helicoid) are richly curved, with K < 0 — 'minimal' refers to area, never to curvature being zero.
The catenoid is the surface you get by spinning a catenary (a hanging-chain curve) around an axis; it is the soap film that spans two parallel circular rings. At every point its two principal curvatures are exactly opposite, k_1 = -k_2, so H = 0 — it is minimal — while K = -k_1^2 < 0, so it is saddle-shaped everywhere. The helicoid (a spiral ramp, like a corkscrew staircase swept out by a horizontal line rotating up an axis) is another minimal surface; remarkably, the catenoid and helicoid can be bent into each other without stretching.
The catenoid: a soap film between two rings, H = 0 yet K < 0 everywhere.
H = 0 means the area is STATIONARY, not necessarily least. A large minimal surface can be unstable — a small perturbation lowers its area — just as a long soap film between two rings collapses when the rings are pulled too far apart. 'Minimal' is a slight misnomer for 'critical point of area.'