On the total mean curvature of non-rigid surfaces

TL;DR

This paper demonstrates that the total mean curvature of a smooth surface in Euclidean 3-space remains stationary under infinitesimal flexes, using Green's theorem to relate curvature variation to a line integral.

Contribution

It provides a concise proof that the total mean curvature is stationary during infinitesimal flexes, connecting curvature variation to a line integral via Green's theorem.

Findings

Total mean curvature is stationary under infinitesimal flexes.

Reduction of curvature variation to a line integral using Green's theorem.

Immediate derivation of a well-known theorem.

Abstract

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.