Global stability of some totally geodesic wave maps

We prove that wave maps that factor as $\mathbb{R}^{1+d} \overset{\varphi_{\text{S}}}{\to} \mathbb{R} \overset{\varphi_{\text{I}}}{\to} M$, subject to a sign condition, are globally nonlinear stable under small compactly supported perturbations when $M$ is a space-form. The main innovation is our assumption on $\varphi_{\text{S}}$, namely that it be a semi-Riemannian submersion. This implies that the background solution has infinite total energy, making this, to the best of our knowledge, the first stability result for factored wave maps with infinite energy backgrounds. We prove that the equations of motion for the perturbation decouple into a nonlinear wave--Klein-Gordon system. We prove global existence for this system and improve on the known regularity assumptions for equations of this type.