Tjyylli

Hyperbolic PDE (like the wave equation) are roughly speaking, PDE that satisfy the “finite propagation speed of information” property. They are ubiquitous in mathematical physics (essentially, most fundamental laws of nature are hyperbolic).

However, do hyperbolic PDE occur in any other areas of mathematics that do not have ties to the real world ?

Hyperbolic PDEs arise unexpectedly in some differential geometric questions involving prescribed data. What's weird in these cases is that there is no natural time coordinate in the PDEs. Here are some examples:

1. Bryant; Griffiths; Yang.
   Characteristics and existence of isometric embeddings.
   Duke Math. J. 50 (1983), no. 4, 893–994.




2. DeTurck, Yang.
   Existence of elastic deformations with prescribed principal strains and triply orthogonal systems.
   Duke Math. J. 51 (1984), no. 2, 243–260. As an aside, the triply orthogonal system result implies the local existence of coordinates on a Riemannian 3-manifold for which the metric tensor is diagonal. This generalizes isothermal coordinates on a Riemannian 2-manifolds.

I have no example of hyperbolic PDE occuring in say, pure, mathematics. Perhaps one deep reason is that the notion of hyperbolic operator distinguishes a convex cone of directions which is inherently a cone of future. Therefore, there is always a distinction between time-like curves and space-like hypersurfaces ; whence the occurence of the real world. In other words, the notion of time and space in inherent to the realm of hyperbolic differential operators.

Nevertheless, the theory of hyperbolic PDEs touches mathematics per se in some places. I have in mind the theory of hyperbolic polynomials, discovered by L. Garding. These are principal symbols of hyperbolic operators. A hyperbolic polynomial $P$ of degree $n$ is positive in the future cone $Gamma$, and the function $P^{frac1n}$ is concave in $Gamma$. An example is $P=det$ in the space of $ntimes n$ symmetric matrices, with $Gamma={bf Sym}_n^+$. The $n$-linear $phi$ form associated with $P$ satisfies the inequality
$$P^{frac1n}(xi_1)cdots P^{frac1n}(xi_n)lephi(xi_1,ldots,xi_n),qquadforall xi_1,ldots,xi_ninGamma.$$
For instance, if $n=2$, this means that the quadratic form $P$ satisfies the converse of Cauchy-Schwarz in the future cone. The polynomial $x_1cdots x_n$ is hyperbolic, its future cone is the first orthant and the corresponding $n$-linear form is nothing but the permanent of a square matrix. The so-called Van der Warden conjecture (now a theorem) is actually a special of a more general problem about hyperbolic polynomials. Through the theory of hyperbolic polynomials, one touches to Real Algebraic Geometry ; this dates back to Petrowsky's school. Actually, O. Oleinik is famous in both PDE and Algebraic Geometry communities.

I should also mention the theory of lacunae for hyperbolic PDEs, which is a problem in Algebraic Topology. See a Bourbaki seminar by M. Atiyah about that.