Mode stability of self-similar wave maps without symmetry in higher dimensions

TL;DR

This paper extends the proof of mode stability for self-similar wave maps from three to higher dimensions, using a novel quasi-solution method with additional parameters.

Contribution

It generalizes mode stability results for wave maps to all dimensions d ≥ 4, employing a new quasi-solution approach with two extra parameters.

Findings

Mode stability established for all d ≥ 4

First application of quasi-solution method with two parameters in this context

Explicit self-similar solutions exhibit finite time blowup in higher dimensions

Abstract

We consider wave maps from $(1+d)$-dimensional Minkowski space into the $d$-sphere. For every $d \geq 3$, there exists an explicit self-similar solution that exhibits finite time blowup. This solution is corotational and its mode stability in the class of corotational functions is known. Recently, Weissenbacher, Koch, and the first author proved mode stability without symmetry assumptions in $d =3$. In this paper we extend this result to all $d \geq 4$. On a technical level, this is the first successful implementation of the quasi-solution method where two additional parameters are present.