Mode stability of blow-up for wave maps in the absence of symmetry

The wave maps equation in three spatial dimensions with a spherical target admits an explicit blow-up solution. Numerical studies suggest this solution captures the generic blow-up behaviour in the backward light cone of the singularity. In this work, we establish the mode stability of this blow-up solution in the backward light cone of the blow-up point without any assumptions on the symmetries of the perturbation. We classify all smooth mode solutions for growth rates $\lambda$ with $\mathrm{Re} \, \lambda \geq 0$ and demonstrate that the blow-up solution is stable up to the mode solutions arising from the symmetry group of the wave maps equation. Our proof relies on a decomposition of the linearised wave maps equation into a tractable system of symmetry-equivariant ordinary differential equations (ODEs), utilising the representation theory of the stabiliser of the blow-up solution. We then use the quasi-solution method of Costin-Donninger-Glogi\'{c} to show the absence of non-zero smooth solutions for the resulting system of ODEs.