  Arnol'd and Kochergin mixing conservative flows on surfaces stand as the main
and almost only natural class of mixing transformations for which higher order
mixing has not been established, nor disproved. Under suitable arithmetic
conditions on their unique rotation vector, of full Lebesgue measure in the
first case and of full Hausdorff dimension in the second, we show that these
flows are mixing of any order. For this, we show that they display a
generalization of the so called Ratner property on slow divergence of nearby
orbits, that implies strong restrictions on their joinings, that in turn yield
higher order mixing. This is the first case in which the Ratner property is
used to prove multiple mixing outside its original context of horocycle flows
and we expect our approach will have further applications.
