The "Nirenberg trick" or "Nirenberg transformation" is the observation^{1} that the semilinear wave equation
\begin{equation}\label{eq:semi}
- \partial^2_{tt} u + \triangle u = (\partial_t u)^2 - |\nabla u|^2
\end{equation}
can be transformed into the *linear wave equation* by noting that
\[ (- \partial^2_{tt} + \triangle) e^{u} = e^{u} \left[ - \partial_{tt}^2 u + \triangle u - (\partial_t u)^2 + |\nabla u|^2\right] .\]
Therefore, if $u$ solves $\eqref{eq:semi}$ with initial data
\begin{equation}
u(0,x) = f(x), \qquad \partial_t u(0,x) = g(x)
\end{equation}
where $f, g$ are both Schwartz functions, then we can solve for $u$ by

- Solving the linear wave equation $\Box v = 0$ with initial data $v(0,x) = e^{f(x)} - 1$ and $\partial_t v(0,x) = e^{f(x)} g(x)$ (note that the data are both in Schwartz class also).
- Reconstructing $u = \ln(v + 1)$.

A few consequences of this procedure are:

- Immediately we see that as long as $v > -1$, the reconstructed function $u$ is smooth.
- Therefore we have automatically
*small data global well-posedness*for the Cauchy problem for $\eqref{eq:semi}$: for all sufficiently small (in various suitable senses) initial data, the corresponding data for $v$ is small, and therefore the solution $v$ to the linear equation have uniform (small) $L^\infty$ bound. - The singularity for $u$, if it occurs, are due to $u \searrow -\infty$. Furthermore, there seems to be a somewhat canonical way of making sense of the solution past the singularity.

The goal of this post is to make sense of the Nirenberg trick from a geometric perspective.

## Wave-maps

Given a Minkowski space $\mathbb{R}^{1,d}$ and a Riemannian manifold $(M,g)$ (where we let $x^i$ with $i = 0, \ldots, d$ be the standard coordinate for $\mathbb{R}^{1,d}$ and $y^A$ for abstract local coordinates for $M$), a *wave-map* $u: \mathbb{R}^{1,d} \to M$ is a formal critical point of the functional
\[ S[u] = \int g(u)_{AB} \partial_i u^A \partial_j u^B m^{ij} ~\mathrm{d}x. \]
The symbol $m^{ij}$ is of course the Minkowski metric.
Its Euler-Lagrange equation takes the form
\begin{equation}\label{eq:genwm}
\Box u^A = \Gamma^A_{BC} \partial_i u^B \partial_j u^C m^{ij}
\end{equation}
where $\Gamma$ is the Christoffel symbols for $(M,g)$ in the coordinates $y^A$.
In the case where $(M,g)$ is the Euclidean space (of $N$ dimensions), the Euler-Lagrange equations can be reduced to $N$ decoupled linear wave equations.

Now let us consider the case where $(M,g)$ is a **one-dimensional** *complete* Riemannian manifold; this means that either $(M,g)$ is the real line $\mathbb{R}$ with the standard metric or its quotient $\mathbb{R} / \lambda \mathbb{Z}$ which is just a circle with circumference $\lambda$.
In the latter case we can equivalently study the wave-map problem by first lifting to the universal cover (which is just $\mathbb{R}$) and then projecting the solution to the circle. Hence we see that in these cases the wave-map equation is identical to the linear wave equation.

However, what if we chose a non-standard coordinate system for $\mathbb{R}$?

Let $I$ be an interval and suppose $y: I \to \mathbb{R}$ is a diffeomorphism to its image. Parametrize $I$ by $z$. The pull-back metric on $I$ is $(y')^2~ \mathrm{d}z^2$. Let's write $\phi = y^{-1}\circ u: \mathbb{R}^{1,d} \to I$. The action functional is then equivalently \[ S[\phi] = \int (y'(\phi))^2 m^{ij} \partial_i \phi \partial_j \phi ~\mathrm{d}x. \] And the corresponding Euler-Lagrange equation is \begin{equation}\label{eq:1dwm} \Box \phi + \frac{y''(\phi)}{y'(\phi)} m^{ij} \partial_i\phi \partial_j\phi = 0. \end{equation}

Observe that $\eqref{eq:semi}$ is merely the special case of $\eqref{eq:1dwm}$ with $y'(\phi) = e^\phi$.

## Semilinear equations

In general, we can reverse the derivation of $\eqref{eq:1dwm}$. Given a wave equation \[ \Box \phi + F(\phi) m^{ij} \partial_i \phi \partial_j\phi = 0 \] we have that if we set \[ y(\phi) = \int_0^\phi \exp\left( \int_0^\sigma F(s) ~\mathrm{d}s \right) ~\mathrm{d}\sigma.\] Then the equation can be written in the form of $\eqref{eq:1dwm}$. Therefore, if we rewrite $u = y\circ \phi$ the function $u$ now must solve a linear wave equation.

The singularity that we can observe for $\phi$ now really is merely due to $y(I)$ possibly not covering the entirety of $\mathbb{R}$. In other words, $I$ with the pull-back metric may be an *incomplete* Riemannian manifold. The singularity is merely just us running to the "end" of the target domain.

## Back to geometry

Returning to the wave-maps equation, one sees that the "best" coordinate system is the one obtained from the arc-length parametrization of the one-dimensional manifold $(M,g)$.
This can be also generalized to the wave-map equation when $M$ is higher dimensional.
Suppose the coordinate system $y^A$ on $M$ is the *geodesic normal coordinate system* based at some point $p\in M$. The geodesic normal coordinate system is, in many ways, the natural generalization of arc-length parametrization in the one-dimensional case; for general manifolds, however, the coordinate system is only local, covering a neighborhood within the injectivity radius at $p$.

Returning to the general equation $\eqref{eq:genwm}$, in the small data regime we know that in geodesic normal coordinates $p$ is associated to the origin, and $\Gamma^A_{BC}(0) = 0$.
Therefore the nominally *quadratically* nonlinear equation $\eqref{eq:genwm}$ can *always* be cast into "normal form" with $\Gamma^A_{BC}(u) = O(u)$, and we thus see that the nonlinearity are in fact at least cubic.
This "normal form" transformation is nothing more than the statement that any Riemannian manifold is *locally* Euclidean.

Incidentally, noting that the Riemannian curvature tensor gives the coefficients of the quadratic Taylor expansion of the metric in geodesic normal coordinates, we see that if $p$ is such that $(M,g)$ has vanishing curvature there, then the corresponding wave-maps equation into its neighborhood will be a *quartic* nonlinear equation.

Now, the normal form implies that the small data Cauchy problem for $\eqref{eq:genwm}$ is automatically globally well-posed when the domain $\mathbb{R}^{1,d}$ has spatial dimension $d\geq 3$.
When $d = 2$ that cubic nonlinear equations have small data global well-posedness needs the null structure; I verified this in a recent paper.
In the $d = 1$ case the small data global well-posedness follows from a easy generalization^{2} of a result of Kevin Luli, Shiwu Yang, and Pin Yu.

- Interestingly, in an illustration of the
*Arnold Principle*at work, this observation seems to be actually due to Klainerman.^{^} - They only stated their result for the case $\Gamma^A_{BC}$ is constant. But the type of preservation of decay that they proved can be used to bootstrap to obtain uniform bounds on $u$ if the initial data has sufficiently fast decay.
^{^}