The energy functional of nonlinear plate theory is a curvature functional for surfaces first proposed on physical grounds by G. Kirchhoff in 1850. We show that it arises as a -limit of three-dimensional nonlinear elasticity theory as the thickness of a plate goes to zero. A key ingredient in the proof is a sharp rigidity estimate for maps v : U → R n , U ⊂ R n . We show that the L 2 -distance of ∇v from a single rotation matrix is bounded by a multiple of the L 2 -distance from the group SO(n) of all rotations.

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