LetXt(x) be a singular Process describing a flow of diffeomorphisms,andkbe a compact surface inwith positive finite Hausdorff measure. We present conditions under which the area ofXt:X(k)goes to zero almost surely and in moments ast→∞. It is shown, in particular, that the flowXtX(·) asymptotically nullifies the arc-length of oriented rectifiable arcs. As a consequence, we recover one of our earlier results on the asymptotic flatness of singular diffusions. Though we work with degenerate diffusions, our method applies to the nondegenerate case as well