Isothermal-related functions in higher dimensions

Question

I am interested in getting some geometrical or analytical perspective in studing the following complex pde. I would appreciate any help.

Consider $ (M,g)$ to be a 3 dimensional Riemannian manifold and consider solving the following pde near a surface $\Sigma$

$<d\Phi,d\Phi>_g =0$ where $\Phi$ is a complex valued function.

If the dimenion is 2 then any function of $z=x_1 + i x_2$ would do the job in isothermal coordinates.

I understand the question is a bit vague but I am interested in general observations about solvability of this pde. Thanks,

Answer 1

In the analytic category, there is a (local) existence theorem, but I don't know about the smooth category. (I rather suspect that there might be some obstructions in the smooth case.) Here is what you can say in the analytic case (and this works in all dimensions):

Suppose that $(M^n,g)$ is a real-analytic Riemannian manifold and that $\Sigma\subset M$ is an embedded real-analytic hypersurface with a choice of normal unit vector field $N$ along $\Sigma$. Denote the induced metric on $\Sigma$ by $\bar g$. Suppose that $f:\Sigma\to \mathbb{C}$ is a real-analytic function that satisfies $\langle \mathrm{d}f,\mathrm{d}f\rangle_{\bar g}= -h^2$ where $h:\Sigma\to\mathbb{C}$ is nonvanishing. Then there exists an open $\Sigma$-neighborhood $U\subset M$ and a function $\Phi:U\to\mathbb{C}$ that satisfies the differential equation $\langle \mathrm{d}\Phi,\mathrm{d}\Phi\rangle_{g}= 0$ and the initial conditions $\mathrm{d}\Phi(N) = h$ and $\Phi = f$ along $\Sigma$. Moreover, the solution is unique in the sense that any other solution $\Phi':U'\to\mathbb{C}$ of the differential equation with these initial conditions that is real-analytic satisfies $\Phi=\Phi'$ on some open $\Sigma$-neighborhood $U''\subset U\cap U'$.

All of this follows as a direct application of the Cartan-Kähler Theorem.

Answer 2

Your solution for $2$ dimensions works for any dimension. You simply solve for co-ordinates $x^1, \dots, x^n$ such that the metric tensor $g_{ij}dx^idx^j$ satisfies $$ g_{11} = g_{22},\ g_{12} = 0 $$ and set $f = x^1$, $g = x^2$. This is effectively parameterized isothermal co-ordinates, so your title is justified. You should be able to get solutions near a point by fixing a foliation of $2$-dimensional surfaces and solving for isothermal co-ordinates along each surface in such a way that the solution depends smoothly on the other $n-2$ co-ordinates.

ADDED: Here's a more explicit description in dimension 3: Start with co-ordinates $y^1$, $y^2$, and $y^3$ (where the level sets of $y^3$ give the foliation.) and a metric $g = g_{ij}dy^idy^j$. We want to solve for functions $\phi^1$ and $\phi^2$ such that if $$ y^1 = \phi^1(x^1,x^2,x^3),\ y^2 = \phi^2(x^1,x^2,x^3),\ y^3 = x^3 $$ then the metric written with respect to $x^1, x^2, x^3$, $g = \hat{g}_{ij}dx^idx^j$ satisfies $$ \hat{g}_{11} = \hat{g}_{22} \text{ and }\hat{g}_{12} = 0. $$ A straightforward calculation shows that this is equivalent to $$ g_{ab}\partial_1\phi^a\partial_1\phi^b = g_{ab}\partial_1\phi^a\partial_1\phi^b\text{ and } g_{ab}\partial_1\phi^a\partial_2\phi^b = 0, $$ where we sum over $1 \le a, b \le 2$. This is the same system of PDE's corresponding to finding co-ordinates $x^1, x^2, x^3$ such that $x^1, x^2$ are isothermal co-ordinates on any level set of $x^3$. In particular, since it involves only differentiation in the $x^1$ and $x^2$ directions and is elliptic, it can be solved for each value of the "parameter" $x^3$ using the standard approach for constructing isothermal co-ordinates. You can then verify that $\Phi = dx^1 + i dx^2$ satisfies $\langle d\Phi,d\Phi\rangle_g = 0$.

CORRECTION: Robert Bryant points out that the wrong problem is being solved above. The correct PDE is $$ 0 = \langle\partial\Phi,\partial\Phi\rangle_g = g^{ij}\partial_i\Phi\partial_j\Phi. $$ Since $\Phi$ is complex-valued, this consists of 2 real equations for two real-valued functions. The linearization can be shown to be elliptic, if $\partial\Phi$ has maximal rank. Therefore, local solutions exist. However, the construction above of isothermal co-ordinates on each leaf of a foliation does not work.

CORRECTION to the CORRECTION: The linearized system is not elliptic. It is, however, generically hypoelliptic subelliptic. Given this, I believe that local solutions exist, but the only way I can see how to prove this is by proving smooth tame estimates for a solution to the hypoelliptic subelliptic PDE and using the Nash-Moser implicit function theorem.