Usefulness of Nash embedding theorem

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.)

In an influential paper, Li and Yau introduced the notion of conformal volume of a Riemannian manifold.

Li, Peter; Yau, Shing-Tung, A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces, Invent. Math. 69, 269-291 (1982). ZBL0503.53042.

See also El Soufi and Ilias for the generalization of the application to first eigenvalues in all dimensions.

For a Riemannian manifold $M$, and $\phi: M\to S^n$ a (branched) conformal immersion, $$V_c(n, \phi) = \sup_{\gamma \in G} V(M,(\gamma\circ\phi)^* can),$$ where $G$ is the group of conformal (Möbius) transformations of $S^n$, and $can$ is the canonical round metric on $S^n$. Then $V_c(n,M) =\underset{ \phi:M\to S^n}{\inf} V_c(n,\phi)$, where the infimum is taken over all conformal immersions into $S^n$. Moreover, $V_c(M)=\lim_{n\to \infty} V_c(n,M)$.

Then $V_c(M)$ is well-defined because of the Nash embedding theorem: there is an isometric embedding of $M$ into $\mathbb{R}^n$ for $n$ sufficiently large, and hence an conformal immersion into $S^n$.

El Soufi and Ilias prove the theorem:

Theorem Let $(M,g)$ be a compact Riemannian manifold of dimension $m$. Then $$\lambda_1(M,g) V(M,g)^{2/m}\leq V_c(M)^{2/m}.$$

Equality holds iff $(M,g)$ admits an isometric immersion into $S^n$ (up to scaling) by first eigenfunctions.

Sobolev mappings between Riemannian manifolds. Let $N$ be closed, and $M$ is compact, possibly with boundary. A natural definition of Sobolev mappings between Riemannian manifolds $W^{1,p}(M,N)$ requires the isometric embedding ot $N$ into an Euclidean space $N\subset\mathbb{R}^\nu$ which is the Nash theorem. Then the space is defined as $$ W^{1,p}(M,N)=\{u\in W^{1,p}(M,\mathbb{R}^\nu):\, u(x)\in N \text{ a.e.}\} $$ The space is equipped with the metric inherited from $W^{1,p}(M,\mathbb{R}^\nu)$.

The space does not depend on the isometric embedding of $N$. If $\iota_1:N\to\mathbb{R}^{\nu_1}$ and $\iota_2:N\to\mathbb{R}^{\nu_2}$ are two isometric embeddings and we denote by
$$ W^{1,p}_{\iota_1}(M,N) \quad \text{and} \quad W^{1,p}_{\iota_2}(M,N) $$ the spaces obtained with respect to these embeddings, then for a mapping $u:M\to N$ we have that $$ \iota_1\circ u\in W^{1,p}_{\iota_1}(M,N) \quad \text{if and only if} \quad \iota_2\circ u\in W^{1,p}_{\iota_2}(M,N). $$ However, the metric in the space depends on the embedding, but the map $$ W^{1,p}_{\iota_1}(M,N)\ni u\mapsto \iota_2\circ\iota_1^{-1}\circ u\in W^{1,p}_{\iota_2}(M,N) $$ is a homeomorphism of spaces.

This class of mappings appear in a natural way in the study of geometric variational problems for mappings between manifolds. Like for example, the theory of harmonic mappings. One of the early problems was the question whether smooth mappings are dense. That led to a very fruitful research showing deep connections to algebraic topology.

You can find some basic information about this space as well as references in the survey paper (available on my website):

P. Hajłasz, Sobolev mappings between manifolds and metric spaces. In: Sobolev Spaces in Mathematics I. Sobolev type Inequalities pp. 185-222. International Mathematical Series. Springer 2009.