Mean Curvature

Differential Geometry · Axiom Academy

Understanding the average curvature of surfaces through principal curvatures, formulas, and minimal surfaces 1. Definition via Principal Curvatures The mean curvature H is defined as the arithmetic average of the two principal curvatures at a point on a surface. The principal curvatures κ₁ and κ₂ represent the maximum and minimum normal curvatures at that point. At each point on a smooth surface, there exist two perpendicular directions (principal directions) along which the normal curvature achieves its extreme values. Mean curvature captures the average bending behavior across these principal directions. Mean curvature can be computed directly from the first and second fundamental forms of the surface. If the first fundamental form has coefficients E, F, G and the second fundamental form has coefficients e, f, g , then: This formula is particularly useful because it relates mean curvature to the shape operator S . Specifically, H = trace(S)/2, where the shape operator encodes how the surface normal changes as we move along the surface. A surface is called minimal if its mean curvature vanishes everywhere: H = 0. This occurs when the principal curvatures are equal in magnitude but opposite in sign (κ₁ = -κ₂), meaning the surface curves equally but in opposite directions. Minimal surfaces naturally arise as soap films: when a wire frame is dipped in soap solution, the resulting film minimizes surface area under the boundary constraint, creating a surface with H = 0.

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