a minimal surface
Bend a closed loop of wire into any shape, dip it in soapy water, and lift it out: the soap film that clings to the wire is a minimal surface. Surface tension pulls the film to use as little area as possible for the boundary it is forced to span, and the resulting shape is exactly the two-dimensional version of a minimal submanifold. These are among the most beautiful objects in geometry — the plane, the catenoid, the helicoid, Scherk's surfaces, Costa's surface — and they are the historical birthplace of geometric analysis.
A minimal surface is a two-dimensional surface with zero mean curvature, H = 0, equivalently a surface whose principal curvatures kappa_1 and kappa_2 satisfy kappa_1 + kappa_2 = 0 at every point (so the surface is locally a balanced saddle). There are several equivalent faces of the definition that are worth knowing: (1) it is area-critical, the Euler-Lagrange surface of the area functional; (2) when written as a graph z = u(x, y) it satisfies the minimal-surface equation; (3) in isothermal (conformal) coordinates its coordinate functions are harmonic, so a minimal surface in R^3 is locally the real part of a holomorphic curve — the Weierstrass-Enneper representation makes every minimal surface out of a holomorphic function and a meromorphic one. This last viewpoint ties minimal surfaces to complex analysis and is why their study is so rich.
Minimal surfaces matter far beyond soap films: they model interfaces minimizing energy, they are test objects for the structure of singular varieties, and existence questions about them (Plateau's problem) launched modern PDE and geometric measure theory. The honest caveat repeats the one for submanifolds: 'minimal' = area-critical, not least-area. A large piece of a catenoid is minimal but no longer minimizing — past a critical size the soap film prefers to snap into two disks. And a minimal surface need not be a graph or even embedded; many of the most interesting examples self-intersect or have ends going off to infinity. Finally, conventions on the sign of mean curvature differ, so 'H = 0' is convention-independent only because it is the zero case.
Two coaxial circular rings dipped together span a catenoid soap film as long as they are close; pull them apart past a critical separation and the catenoid becomes unstable and the film jumps to two flat disks — the minimal surface still exists mathematically but is no longer the minimizer.
The catenoid between two rings: minimal yet only minimizing below a critical ring separation.
The Weierstrass-Enneper representation builds minimal surfaces in R^3 from holomorphic data, but it is special to flat R^3; in a curved ambient manifold there is no such complex-analytic shortcut, and the equation must be solved as a genuine nonlinear PDE.