In the complex plane, there is a correspondence between solutions of the Monge?Ampère equation and solutions of a certain differential inclusion associated to SO(2). Under this correspondence, the W^{2,1+epsilon} regularity of solutions to Monge?Ampère is rephrased as a quantitative unique continuation principle for solutions of the differential inclusion. We will sketch a proof of the latter, which relies on quasiconformal maps and the rigidity estimate for SO(2). Based on joint work with G. De Philippis and R. Tione