# Global stability of some totally geodesic wave maps

## TL;DR

This paper establishes the global nonlinear stability of certain wave maps factoring through a semi-Riemannian submersion into space-forms, even with infinite energy backgrounds, by analyzing a coupled wave--Klein-Gordon system.

## Contribution

It introduces a novel stability result for factored wave maps with infinite energy backgrounds, assuming the semi-Riemannian submersion condition on the background map.

## Key findings

- Proves global stability of factored wave maps under small perturbations.
- Shows the perturbation equations decouple into a wave--Klein-Gordon system.
- Improves regularity assumptions for analyzing such systems.

## Abstract

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.