Entropy, elasticity, and the isometric embedding problem: M³ → ℝ⁶

The balance laws for isometric embedding of a 3-dimensional Riemannian manifold into the 6-dimensional Euclidean space are exposited. It is shown that a necessary and sufficient condition for solving the equations of these balance laws is a system reminiscent of the equations of elastostatics. In turn, this elastostatic system possesses an elementary entropy equation in the sense of Lax. In addition, we use an entropy equality for the linearized system (linearized about a known embedding) to develop L 2 estimates for the dependent variables. © Springer-Verlag Berlin Heidelberg 2014.