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.
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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: