The Nash embedding theorem is an existence theorem for a certain nonlinear PDE ($\partial_i u \cdot \partial_j u = g_{ij}$) and it can in turn be used to construct solutions to other nonlinear PDE. For instance, in my paper

*Tao, Terence*, **Finite-time blowup for a supercritical defocusing nonlinear wave system**, Anal. PDE 9, No. 8, 1999-2030 (2016). ZBL1365.35111.

I used the Nash embedding theorem to construct discretely self-similar solutions to a supercritical defocusing nonlinear wave equation $-\partial_{tt} u + \Delta u = (\nabla F)(u)$ on (a backwards light cone in) ${\bf R}^{3+1}$ that blew up in finite time. Roughly speaking, the idea was to first construct the stress-energy tensor $T_{\alpha \beta}$ and then find a field $u$ that exhibited that stress-energy tensor; the stress-energy tensor $T_{\alpha \beta} = \partial_\alpha u \cdot \partial_\beta u -\frac{1}{2} \eta_{\alpha \beta} ( \partial^\gamma u \cdot \partial_\gamma u + F(u))$ was close enough to the quadratic form $\partial_i u \cdot \partial_j u$ that shows up in the isometric embedding problem that I was able to use the Nash embedding theorem (applied to a backwards light cone, quotiented by a discrete scaling symmetry) to resolve the second step of the argument. The field $u$ had to take values in quite a high dimensional space - I ended up using ${\bf R}^{40}$ - because of the somewhat high dimension needed in the target Euclidean space for the Nash embedding theorem to apply.

Also, there is a major indirect use of the Nash embedding theorem: the Nash-Moser iteration scheme that was introduced in order to prove this theorem has since proven to be a powerful tool to establish existence theorems for several other nonlinear PDE, though in many cases it turns out later that with some trickery one can avoid this scheme. For instance the original proof by Hamilton of the local existence for Ricci flow in

*Hamilton, Richard S.*, **Three-manifolds with positive Ricci curvature**, J. Differ. Geom. 17, 255-306 (1982). ZBL0504.53034.

relied on Nash-Moser iteration, though a later trick of de Turck in

*DeTurck, Dennis M.*, **Deforming metrics in the direction of their Ricci tensors**, J. Differ. Geom. 18, 157-162 (1983). ZBL0517.53044.

allowed one to avoid using this scheme. (For Nash embedding itself, a somewhat similar trick of Gunther in

*Günther, Matthias*, Isometric embeddings of Riemannian manifolds, Proc. Int. Congr. Math., Kyoto/Japan 1990, Vol. II, 1137-1143 (1991). ZBL0745.53031.

can be used to also avoid applying Nash-Moser iteration.)

isometricallyembedded? Can we really not just use any smooth embedding into $\mathbb R^N$? $\endgroup$ – John Pardon Oct 6 '19 at 21:321more comments