A remark on isolated removable singularity of harmonic maps in dimension two

For a ball $B_R(0)\subset {\mathbb{R}}^2$, we provide sufficient conditions such that a harmonic map $u\in C^{\mathrm{\infty }}(B_R(0)\setminus \{0\},N)$, with a self-similar bound on its gradient, belongs to $C^{\mathrm{\infty }}(B_R(0))$. Those conditions also guarantee the triviality of such harmonic maps when $R=\mathrm{\infty }$.