Wave maps from Gödelʼs universe

Using a result by L. Koch we realize G\"ode's universe $\mathcal{G}^4_\alpha = (\mathbb{R}^4, g_\alpha )$ as the total space of a principal $\mathbb{R}$-bundle over a strictly pseudoconvex CR manifold $M^3$ and exploit the analogy between $g_\alpha$ and Fefferman's metric $F_\theta$ to show that for any $\mathbb{R}$-invariant wave map $\Phi$ of $\mathcal{G}^4_\alpha$ into a Riemannian manifold $N$, the corresponding base map $\phi : M^3 \to N$ is subelliptic harmonic, with respect to a canonical choice of contact form $\theta$ on $M^3$. We show that the subelliptic Jacobi operator $J^\phi_b$ of $\phi$ has a discrete Dirichlet spectrum on any bounded domain $D \subset M^3$ supporting the Poincar\'e inequality on $\IN{W}^{1,2}_H (D, \phi^{-1} T N)$ and Kondrakov compactness i.e. compactness of the embedding $\IN{W}^{1,2}_H (D, \phi^{-1} T N) \hookrightarrow L^2 (D, \phi^{-1} T N)$. We exhibit an explicit solution $\pi:\mathcal{G}^4_\alpha \to M^3$ to the wave map system on $\mathcal{G}^4_\alpha$, of index $\mathrm{ind}^\Omega (\pi ) \geq 1$ for any bounded domain $\Omega \subset \mathcal{G}^4_\alpha$. Mounoud's distance $d^\infty_{G_0, \Omega} (g_\alpha, F_\theta )$ is bounded below by a constant depending only on the rotation frequency of G\"odel's universe, thus giving a measure of the bias of $g_\alpha$ from being Fefferman like in the region $\Omega \subset \mathbb{R}^4$.