On the total mean curvature of non-rigid surfaces

Using Green’s theorem we reduce the variation of the total mean curvature of a smooth surface in the Euclidean 3-space to a line integral of a special vector field and obtain the following well-known theorem as an immediate consequence: the total mean curvature of a closed smooth surface in the Euclidean 3-space is stationary under an infinitesimal flex.

💡 Research Summary

The paper investigates how the total mean curvature of a smooth surface embedded in Euclidean three‑space changes under an infinitesimal flex, i.e., a first‑order isometric deformation that is not generated by a rigid motion. After clarifying the notions of a “flexible” surface (a one‑parameter family of isometric surfaces that are not congruent for different parameters) and a “non‑rigid” surface (a surface admitting a non‑trivial infinitesimal flex), the author derives a compact formula for the variation of the total mean curvature.

Let S⊂ℝ³ be a compact oriented smooth surface, and let v:S→ℝ³ be a vector field defining an infinitesimal flex via the map ψ(x,t)=x+t v(x). For each t the deformed surface ψ(S,t) has a unit normal n(x,t); its time derivative at t=0 is denoted n′(x,0). The author introduces the tangential vector field
m(x)=n′(x,0)×n(x,0)
on S. The total mean curvature is defined in the classical way
H(S)=∫S ½(κ₁+κ₂) dS,
and its first variation under the flex is
H′(S)=d/dt|{t=0} H(ψ(S,t)).

Using a local graph parametrisation x(u,v)=(u,v,f(u,v)) and the infinitesimal flex conditions (which translate into a system of linear PDEs for the components ξ,η,ζ of v), the author computes H′(S) explicitly. The computation yields
2 H′(S)=∬_D