Local wellposedness for the 2+1 dimensional monopole equation

The space-time monopole equation on ${\mathbb{R}}^{2+1}$ can be derived by a dimensional reduction of the anti-self-dual Yang Mills equations on ${\mathbb{R}}^{2+2}$. It can be also viewed as the hyperbolic analog of Bogomolny equations. We uncover null forms in the nonlinearities and employ optimal bilinear estimates in the framework of Wave-Sobolev spaces. As a result, we show the equation is locally wellposed in the Coulomb gauge for initial data sufficiently small in $H^s$ for $s>1\mathrm{/}4$.