a minimal submanifold

If a geodesic is the higher-dimensional cousin of a straight line — the shortest path — a minimal submanifold is the cousin of a flat plane: a surface (or higher-dimensional sheet) that cannot reduce its area by any small wiggle. Dip a wire loop in soapy water and the film that spans it pulls itself as tight as it can; that taut film is the everyday picture. The word 'minimal' is a historical label and slightly misleading: it really means area-critical, the analogue of 'flat spot' rather than guaranteed least area.

Precisely, an immersed submanifold Sigma of a Riemannian manifold (M, g) is minimal when its mean curvature vector vanishes identically, H = 0. The mean curvature is the trace of the second fundamental form, H = trace(II) = sum_i II(e_i, e_i) over an orthonormal frame; it measures the average bending of Sigma inside M. By the first variation of area, d(Area)/ds at s=0 = -integral over Sigma of g(V, H) dA for any compactly supported normal variation V, so H = 0 is exactly the Euler-Lagrange equation 'area is stationary'. Equivalently, the coordinate functions of M restricted to Sigma are harmonic with respect to the induced Laplacian, which is why minimal submanifolds and harmonic-function theory are so intertwined. A plane in R^3 is minimal (it does not bend at all); the catenoid and helicoid are the classical nontrivial minimal surfaces; a great sphere S^{k} inside the round S^{n} is a minimal submanifold.

Minimal submanifolds sit at the crossroads of geometry, PDE, and the calculus of variations, and they power deep theorems — the positive mass theorem, rigidity results, and the structure of singularities all use them. The crucial honesty point is the gap between 'minimal' and 'minimizing': H = 0 is only the first-variation (stationary) condition. Whether a minimal submanifold is stable (positive second variation) or actually area-minimizing is a strictly stronger question governed by the Jacobi/stability operator; the equator of a sphere is minimal and even stable, but a great two-sphere's complement shows unstable minimal surfaces are common. Also, in dimension and codimension large enough, area-minimizers can have singularities (the Simons cone), so smoothness is not automatic.