The $\bar{\mathrm{\partial }}$-Neumann problem and boundary integral equations

In this note, we find an equivalent boundary integral equation to the classical $\bar{\mathrm{\partial }}$-Neumann problem. The new equation contains an equivalent regularity to the global regularity of the $\bar{\mathrm{\partial }}$-Neumann problem. We also use the integral equations to observe a new-found rigidity property of the $\bar{\mathrm{\partial }}$-Neumann problem on the unit ball in ${\mathbb{C}}^2$.