Studies of Charmonium at BESIII

Based on $\psi (3686)$ decays of 106 million, the 1.31 billion J/psi events and a data sample of $\psi (3770)$ with $2.9f{b}^{-1}$ integrated luminosity, many analyses are performed. Exclusively baryonic decays of the $\psi (3770)$, the radiative transition $\psi (3770)\to \gamma {\eta }_c(2S)$, the $\psi (3770)$ transition to ${\chi }_{cJ}$, isospin violation decay ${\chi }_{c0,2}\to {\pi }^0{\eta }_c$, the $C-$parity violation decays $J\mathrm{/}\psi \to \gamma \gamma ,\gamma \varphi$ are searched for, but no significant signals are observed, and upper limits are set for these decays. The decays of $\psi (3770)\to \gamma {\chi }_{c1}$ and $J\mathrm{/}\psi \to {\pi }^0\varphi$ signals are observed. These measurements provide more information on the charmonium structure, and the isospin and $C$-parity violation in the charmonium decays.