The mean curvature vector of an -dimensional embedded submanifold is the trace . For a surface in , the scalar mean curvature is half the trace of the shape operator:
A minimal submanifold has vanishing mean curvature vector. For a surface in , this says that the scalar mean curvature is zero, equivalently that its two principal curvatures sum to zero.
An outer area-minimizing surface is a closed surface in a Riemannian initial-data slice whose area is no greater than that of any enclosing competitor. This is a variational property, not a consequence of containment alone. It is important when choosing the horizon-area quantity in the Riemannian Penrose inequality and when examining an apparent-horizon area comparison.
The isothermal coordinates give the immersion . Its coordinate functions are harmonic and its metric is , so it is a complete immersed minimal surface with self-intersections.
With and , a variation with velocity satisfiesThus a boundaryless submanifold is stationary for every compactly supported variation exactly when it is minimal.
For the squared distance , a minimal -submanifold satisfies . On a compact boundaryless manifold, attains a maximum where the Laplacian is nonpositive, a contradiction for .
Replacing the chosen unit normal by replaces the shape operator by . On a surface, Gaussian curvature is unchanged, while mean curvature changes sign.
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Mean curvature is a geometric concept that arises in differential geometry, particularly in the study of surfaces. It measures the average curvature of a surface at a given point and is an important characteristic in the study of minimal surfaces and the geometry of manifolds. For a surface defined in three-dimensional space, the mean curvature \( H \) at a point is given by the average of the principal curvatures \( k_1 \) and \( k_2 \) at that point.