Minimal Submanifold

A minimal submanifold of a Riemannian manifold is an immersed submanifold whose mean curvature vector vanishes. It is a stationary point of volume under compactly supported variations of the immersion. Stationarity does not require local or global volume minimization. A minimal surface is the two-dimensional hypersurface case in a three-dimensional ambient manifold. Totally geodesic submanifolds are minimal, while the minimal Clifford hypersurfaces in a round sphere have nonzero second fundamental form and are also minimal. The Solomon-Yau least-volume conjecture compares closed minimal hypersurfaces in spheres with these products.