Generic ill-posedness for Schr\"odinger equation with power-type nonlinearity on $\mathbb{S}^2$

In this article, we investigate the local well-posedness of the nonlinear Schrödinger equation on the two-dimensional sphere $\mathbb{S}^2$: \begin{align*} i\partial_tu+\Delta_{g}u=F(u). \end{align*} The nonlinearity $F(u)$ is assumed to be gauge-invariant. More presicely, there exists a function $V\in C^\infty(\mathbb{C},\mathbb{R})$ such that $F=\frac{\partial V}{\partial \bar{z}}$. Moreover, $V(z)$ obeys \begin{equation}\label{H-11}
V(e^{i\theta}z)=V(z),\,\,\theta\in\Bbb R,\,\,z\in\Bbb C,
|\partial_z^{k_1}\partial_{\bar{z}}^{k_2}V(z)|\leq C_{k_1,k_2}(1+|z|)^{1+\alpha-k_1-k_2},\tag{H-1}
\end{equation} for some $\alpha\geq3.$ The main contribution of this paper is the new lower bound of threshold of local well-posedness $s_c(\mathbb{S}^2,\alpha)$. Specifically, under assumption \eqref{H-11}, we prove that for $\alpha \geq 3$, the equation is ill-posed in $H^s(\mathbb{S}^2)$ with $s < 1 - \frac{2}{\alpha-1}$ in the sense that the norm inflation occurs. Combined with the well-posedness in Yang [Sci. China Math. 58 (2015), 1023-1046], the exact threshold $s_c(\mathbb{S}^2,\alpha)$ for $\alpha\geq5$ is $1-\frac{2}{\alpha-1}$, which matches the scaling-critical regularity as the Euclidean setting. Moreover, for $\alpha \in [3, \frac{11}{3})$, we show that the solution map is not uniformly continuous in the range $0 < s < \frac14$ for the power-type nonlinearity $F(u)=|u|^{\alpha-1}u$, which lies strictly above the scaling-invariant threshold. This provides a new characterization of the ill-posedness regime for all $\alpha \geq 3$, extending an earlier result of Burq-Gérard-Tzvetkov [Math. Res. Lett. 9 (2002), 323-335]. Our result can also be regarded as a Schrödinger counterpart of Xia [Int. Math. Res. Not. (2021), 15533-15554].