Characterizing Two-Dimensional Maps Whose Jacobians Have Constant Eigenvalues

http://dx.doi.org/10.4153/CMB-2003-034-4
Canad. Math. Bull. 46(2003), 323-331

  Published:2003-09-01
 Printed: Sep 2003

Recent papers have shown that $C^1$ maps $F\colon \mathbb{R}^2 \rightarrow \mathbb{R}^2$ whose Jacobians have constant eigenvalues can be completely characterized if either the eigenvalues are equal or $F$ is a polynomial. Specifically, $F=(u,v)$ must take the form \begin{gather*} u = ax + by + \beta \phi(\alpha x + \beta y) + e \\ v = cx + dy - \alpha \phi(\alpha x + \beta y) + f \end{gather*} for some constants $a$, $b$, $c$, $d$, $e$, $f$, $\alpha$, $\beta$ and a $C^1$ function $\phi$ in one variable. If, in addition, the function $\phi$ is not affine, then \begin{equation} \alpha\beta (d-a) + b\alpha^2 - c\beta^2 = 0. \end{equation} This paper shows how these theorems cannot be extended by constructing a real-analytic map whose Jacobian eigenvalues are $\pm 1/2$ and does not fit the previous form. This example is also used to construct non-obvious solutions to nonlinear PDEs, including the Monge--Amp\`ere equation.