Construction of infinite time bubble tower solutions for energy critical equivariant Schrödinger maps

We prove the existence of infinite time bubble tower solutions for energy critical $k$-equivariant Schrödinger maps $\mathbb{R}^2\to \mathbb{S}^2$ with $k\geq 3$. More precisely, for any integers $k\geq 3$ and $N\geq 1$, we construct a $k$-equivariant solution that is global in one time direction an