Norm inflation for the cubic hyperbolic NLS on ${\mathbb{T}}^2$

We prove norm inflation for the cubic hyperbolic nonlinear Schrödinger equation in $H^s({\mathbb{T}}^2)$ for every $s\in (-\mathrm{\infty },0)\cup (0,\frac{1}{2}]$. The scaling-critical point $s=0$ is excluded by conservation of the $L^2$ norm. The strong ill-posedness below and above the scaling-critical point arises from two completely different mechanisms. Particularly in the scaling-subcritical regime, this dynamical instability stems from the hyperbolic nature. Together with the local well-posedness result in \cite{WangHNLS}, this gives a sharp dichotomy away from the mass space $L^2({\mathbb{T}}^2)$: local well-posedness holds for $s>\frac{1}{2}$, whereas norm inflation occurs for all $s\le \frac{1}{2}$ with $s\mathrm{\ne }0$.