The Nash–Kuiper Theorem and the Onsager Conjecture

We give an account of the analogies between the Nash– Kuiper C solutions of the isometric embedding problem and the weak solutions of the incompressible Euler equations which violate the energy conservation. Such analogies have lead to the recent resolution of a well-known conjecture of Lars Onsager in the theory of fully developed turbulence. 1. The Nash-Kuiper Theorem Let (Σn, g) be a smooth n-dimensional Riemannian manifold. A map u : Σ→ RN is isometric if it preserves the length of curves, i.e. if `g(γ) = `e(u ◦ γ) for any C1 curve γ ⊂ Σ, (1) where `g(γ) denotes the length of γ with respect to the metric g: `g(γ) = ˆ √ g(γ(t))[γ̇(t), γ̇(t)] dt . (2) As customary, in local coordinates we can express the metric tensor g as g = gijdxi ⊗ dxj . For a C1 map u, condition (1) is equivalent to the system of partial differential equations ∂iu · ∂ju = gij . (3) In the usual language of Riemannian geometry, (3) means that g is the pullback of the Euclidean metric through the map u. The existence of isometric immersions (resp. embeddings) of Riemannian manifolds into some Euclidean space is a classical problem, explicitly formulated for the first time by Schläfli, see [46]: in the latter Schläfli conjectured that the system is solvable locally if the dimension N of the target is at least sn := n(n+1) 2 . Such conjecture stands to reason because (3) consists precisely of sn equations in N unknowns. In the first half of the twentieth 1991 Mathematics Subject Classification. 35Q31,(35A01,35D30,53A99,53C21,76F02).