On The 1+2 Dimensional Landau-Lifshitz Equation

By using the geometric concept of PDEs with prescribed curvature representations, we show that the 1+2 dimensional Landau-Lifshitz equation is gauge equivalent to a 1+2 dimensional nonlinear Schr"odinger-type system. From the nonlinear Schr"odinger-type system, we construct blowing up $H^3({\mathbf{R}}^2)$-solutions to the 1+2 dimensional Landau-Lifshitz equation, which reveals the blow up phenomenon of the equation.